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THE  LIBRARY 

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THE  UNIVERSITY 

OF  CALIFORNIA 

LOS  ANGELES 


GIFT  OF 

Johr  S.Proll 


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THE 


ELEMENTS  OF  MECHANICS, 

COMPREHENDING 

STATICS   AND  DYNAMICS. 

WITH 
A  COPIOCB  COLLECTION  OF  ' 

MECHANICAL  PROBLEMS. 

INTENDED  FOR  THE  DSE  OF 

MATHEMATICAL   STUDENTS  IN  SCHOOLS  AND  UNIVERSITIES. 
WITH    NUMEROUS    PLATES. 


BY   J.  R.  YOUNG, 


▲CTROR    OF    "THE    ELEMENTS   OF    ANALYTICAL   GEOMETRY;"    "ELEMENTS    OF    THE 
DiFFERENTIAt    AND  INTEGRAL    CALCULUS." 


REVISED  AND  CORRECTED 


BY  JOHN   D.  WILLIAMS, 

ACTHOR     OF    "KEY    TO   BUTTON'S     M  A  TH  E  M  ATI  C  S,"  &C. 


PHILADELPHIA : 
HOGAN    AND    THOMPSON, 

No.  30,  NORTH  FOURTH  STREET. 

1839. 

JOHM  S.  PRELL 

Gvil  &  Mechanical  Engineer. 

SAN  FRANCISCO,  CAL. 


Entered,  accordincr  to  the  Act  of  Congress,  in  the  year  1834,  by 
Carky,  Lea  &  Blanchard,  in  the  Clerk's  Office  of  the  District  Court  of 
the  Eastern  District  of  Pennsylvania. 


STEREOTYPED  BY   U  JOHNSON, 
miLADELFHIA. 


•J3fiq  .2  VIHOl 


Ulrary 

PREFACE.        i^3l 


The  following  Treatise  is  an  attempt  to  exhibit,  in  small 
compass,  the  principles  of  Mechanical  Science  in  its  present 
improved  state,  and  to  supply  the  English  student  with  a 
clear  and  comprehensive  manual  of  instruction  on  this  im- 
portant branch  of  Natural  Philosophy. 

Our  language  already  possesses  some  very  valuable  works 
in  this  department  of  science,  as  for  instance,  the  treatises  of 
Professors  Gregory  and  Whewell  ;  works  Avhich,  for  the  abun- 
dance of  real  information  that  they  convey,  are  not,  perhaps, 
to  be  equalled  by  any  similar  performances  of  our  continental 
neighbours. 

The  bulk  and  consequent  high  price,  however,  of  these 
works  must  necessarily  place  them  beyond  the  reach  of  many 
students  desirous  to  be  informed  on  the  subjects  of  which  they 
treat ;  and  there  can,  I  think,  be  no  doubt  that  at  a  time  like 
the  present,  when  a  taste  for  analytical  science  is  so  widely 
extending  itself,  a  treatise,  of  moderate  price,  on  Analytical 
Mechanics,  if  well  executed,  would  prove  acceptable  both  to 
teachers  and  to  students. 

Under  these  impressions  I  have  been  led  to  undertake  this 
elementary  Treatise,  with  the  hope  that  by  economizing  the  pa- 
per, and  adopting  a  small  clear  type.  I  might  be  able  to  compress 
into  one  small  volume  a  course  of  instruction  on  elementaiy 
mechanics,  of  extent  amply  sufficient  for  all  the  purposes  of 
academical  education.  My  desire,  however,  having  been  to 
teach  the  elements  of  the  science,  not  to  write  a  book,  I  have 
spared  no  pains  to  render  the  whole  clear  and  intelligible  ;  to 
develope  the  several  theories  with  as  much  simplicity  as  I 
could  ;  to  explain  fully  the  meaning  and  extent  of  the  various 
analytical  expressions  in  which  these  theories  are  embodied  ; 
and  finally,  to  illustrate  each  by  a  sufficient  number  of  useful 
and  interesting  practical  examples. 

To  what  extent  these  endeavours  may  have  been  successful, 
it  is  for  others  now  to  determine;  but,  from  the  very  flattering 
reception  which  my  former  mathematical  publications  have 
met  with,  both  in  this  country  and   in  America,  I  am  encou- 

3 


7137S5 


i  PREFACE. 

raged  to  hope  tlial  the  present  volume  will  be  found,  upon  ex- 
amination, not  altogether  undeserving  of  notice. 

The  work  is  divided  into  two  principal  divisions,  Statics 
or  the  theory  of  Equilibrium,  and  Dynamics  or  the  theory  of 
Motion  ;  and  these  are  again  subdivided  into  sections  and 
chapters.  A  very  short  account  of  these  will  suffice  here,  as 
a  copious  analysis  is  presented  in  the  table  of  contents. 

The  first  section  of  the  Statics  treats  on  the  equilibrium  of  a 
point,  viewed  under  two  aspects  :  first,  as  entirely  free  ;  and 
second,  when  constrained  to  rest  on  a  given  curve  or  surface. 
Into  this  section  too  is  introduced  the  theory  of  the  funicular 
polygon  and  catenary:  this,  I  am  aware,  is  not  in  strict  accord- 
ance with  a  scientific  arrangement  of  the  parts  of  the  subject, 
nor  do  I  consider  such  arrangement  to  be  absolutely  essential  in 
an  elementary  treatise  ;  the  principle  which  with  me  has  all 
along  governed  the  arrangement  is  this,  viz.,  so  to  dispose  the 
several  topics  that  each  may  present  itself  to  the  student  pre- 
cisely at  that  place  where  he  is  best  prepared  to  receive  it, 
and  thus  the  acquisition  of  the  whole  be  facilitated. 

The  second  section  is  on  the  theory  of  the  equilibrium  of 
a  rigid  body  delivered  in  all  its  generality,  and  applied  to  a 
variety  of  examples.  The  closing  ciiapter  of  this  section  is 
devoted  to  a  subject  of  considerable  practical  importance,  the 
strength  and  stress  of  beams  ;  for  the  principal  materials  of 
it  I  am  indebted  to  Mr.  Barlow's  experimental  inquiries  on 
this  subject. 

These  two  sections  comprise  the  theory  of  Statics  ;  the  se-~^ 
cond  part,  or  Dynamics,  is  divided  into  three  sections  ;  the 
first  being  on   the  rectilinear  motion  of  a  free  point,  the  se- 
cond on  its  curvilinear  motion,  and   the  third   discussing  the 
general  theory  of  the  motion  of  a  solid  body. 

In  the  opening  chapter  of  tlie  first  section,  the  fundamental 
equations  of  motion  are  deduced  from  simple  and  obvious  con- 
siderations, and  pains  are  taken  to  give  a  clear  and  distinct 
meaning  to  the  several  analytical  expressions  involved  in 
these  fundamental  equations.  All  these  are  then  fully  illus- 
trated by  interesting  practical  exercises. 

In  the  second  section  a  pretty  comprehensive  view  is  taken 
of  the  theory  of  curvilinear  motion  ;  and  some  attempts  have 
been  made  to  simplify  those  p^irts  of  it  which  seemed  most  to 
require  simplification. 

The  last  section,  on  the  motion  of  a  solid  body,  is  the  most 
A  2 


PREFACE.  5 

extensive,  as  well  as  the  most  difficult,  and  will  be  found  to 
embrace  a  great  variety  of  important  particulars,  treated,  I  hope, 
with  sufficient  clearness  to  be  abundantly  intelligible  to  an 
ordinary  student ;  several  interesting  dynamical  problems  are 
interspersed  throughout  this  section,  and  to  the  end  is  append- 
ed a  miscellaneous  collection,  as  further  illustrative  of  the  use 
and  application  of  the  general  theories  before  established, 
especially  o{  the  Principle  of  D'Alemhert^ow  account  of  the 
importance  of  this  principle  in  a  great  variety  of  dynamical 
inquiries. 

For  the  manner  in  which  the  subjects  here  briefly  enume- 
rated are  discussed,  I  must  now  refer  to  the  book  itself ;  and 
shall  be  glad  if  it  be  thought  calculated  to  promote,  in  any  de- 
gree, the  study  of  the  science,  or  to  form  a  useful  introduction 
to  works  of  higher  pretensions  and  of  acknowledged  ability. 

J.  R.  YOUNG. 
April  10,  1832. 


CONTENTS. 


PART  I. 

ELEMENTS  OF  STATICS. 

Introduction. 

Article  Page 

1.  MECHASfics  defined               -            -            -            -            -  -13 

2.  Force  defined             -            -            -            -            -            -  -     ib. 

3.  Equilibrium  of  a  point  acted  on  by  two  equal  and  opposite  forces  -     ib. 

4.  Linear  measure  of  force          -             -             -             -             -  -     14 

5.  The  point  of  application  of  a  force  may  be  anywhere  in  the  line  of  its 

direction      -  -  -  -  -  -  -  -ib. 


On  the  Equilibrium  of  a  Point, 

6.  The  resultant  of  forces  acting  in  the  same  straight  line  is  equal  to  the 

algebraical  sum  of  those  forces         -  -  -  -  -     17 

7.  The  resultant  of  several  forces  in  one  plane  is  in  that  plane     -  -     18 

8.  When  the  intensities  of  equilibrating  forces  vary,  the  resultant  varies 

in  the  same  ratio,  but  retains  its  direction     -  -  -  -     ib. 

9.  If  three  equal  forces  are  inclined  to  one  another  in  angles  each  of 

120°,  any  one  will  balance  the  other  two      -  -  -  -     ib. 

10.  Two  of  these  forces  are  equivalent  to  a  third,  represented  in  direction 

and  intensity  by  the  diagonal  of  the  rhombus  constructed  on  the 
lines  which  represent  the  forces        -  -  -  -  -     ib. 

11.  Any  two  equal  forces,  represented  by  the  sides  of  a  rhombus,  are  equiva- 

lent to  the  force  represented  by  the  diagonal  -  -  -     19 

12.  Any  two  forces  represented  by  the  sides  of  a  rectangle  are  equivalent  to 

the  single  force  represented  by  the  diagonal         -  -  -     20 

13.  Any  two  forces  represented  by  the  sides  of  a  parallelogram  are  equiva- 

lent to  the  single  force  represented  by  the  diagonal    -  -  -     21 

14.  Of  three  equilibrating  forces  the  intensity  of  any  one  is  proportional  to 

the  sine  of  the  angle  between  the  other  two    -         -  -  -     22 

15.  Composition  of  several  concurring  forces  by  geometrical  construction  -     ib. 

16.  To  resolve  a  single  force  into  two  concurring  forces  acting  in  given  direc- 

tions -  -  -  -  -  -  -  -     23 

17.  Analytical  determination  of  the  intensity  of  the  resultant  of  a  system  of 

concurring  forces  situated  in  one  plane        -  -  -  -     ib. 

18.  Remarks  on  the  signs  of  the  forces     -  -  -  -  -     24 

19.  Determination  of  the  direction  of  the  resultant  of  a  system  of  concurring 

forces  -  -  -  -  -  -  -  -ib. 

20.  Particular  example  of  the  composition  of  forces  -  -  -     25 

21.  On  the  parallelopiped  of  forces  -  -  -  -  -     ib. 

22.  Determination  of  the  resultant  of  any  system  of  concurring  forces  situ- 

ated in  space  -  -  -  -  -  -  -27 

—  Equations  of  equilibrium  of  a  system  of  concurring  forces  -    \b 

n 


8  CONTENTS. 

.IrlicU  Pag' 

23.  When  any  number  of  forces  are  in  equilibrium,  their  projections  upon 

any  plane  or  line  will  also  be  in  equilibrium     -        -             -             -  28 

—  Equation  of  the  line  representing  the  resultant  of  a  system  of  forces     -  ib 

24.  Problems  respecting  the  action  of  forces  through   the  intervention  of 

flexible   cords         ...              .             .              •             -  ib. 

2.5.  On  the  funicular  polygon      -            -             -            -             -            -  34 

26.  The  polygon  formed  by  the  vertical  action  of  weights             -             -  36 

27.  On   the  catenary  curve          -             -             -             -             -             -  37 

28.  Equations  of  the  cur%'e            -             -             -             -             -             -  38 

29.  Determination  of  the  tension  and  direction  of  the  catenary  at  any  point 

when  the  points  of  suspension  are  in  a  horizontal   line         -             -  40 

30.  Forms  of  the  equations  of  the  curve,  and  of  the  tension,  when  the 

origin  is  at  lowest  point  of  the  catenary        -             -             -             -  ib 

31.  Problems  on  suspended  chains          -             -             -             -             -  41 

32.  General  theory  of  the  equilibrium  of  a  point  on  a  surface         -             -  45 

33.  Conditions  of  equilibrium  when  the  acting  force  is  gravity      -             -  46 

34.  Problems  on  the  equilibrium  of  bodies  of  surfaces        -            -             -  48 

SECTION    II. 

On  the  Eqtdlibrinm  of  a  Solid  Body. 

'■\b.  Preliminary  remarks                -             -             -             -             -             -  53 

36.  Determination  of  the  resultant  of  two  parallel  forces    -             -             -  ib 

37.  Determination  of  the  resultant  of  a  system  of  parallel  forces     -             -  55 

35.  Determination  of  the  centre  of  parallel  forces                -             -             -  ib. 

—  Conditions  of  equilibrium  of  a  system  of  parallel  forces            -  57 

39.  The  same  conditions  more  concisely  expressed             -             -  ib. 

40.  On  the  centre  of  gravity          -             -             -             -             -             -  58 

41.  Analytical  expressions  for  the  co-ordinates  of  the  centre  of  gravity         -  60 

42.  Determination  of  the  centre  of  gravity  of  a  plane  line               -             -  62 

43.  Centre  of  gravity  of  a  plane  area         -             -             -             -             -  64 

44.  Centre  of  gra^^ty  of  a  surface  of  revolution       -             -             -             -  66 

45.  Centre  of  gravity  of  a  solid  of  revolution         -             -             -             -  67 

46.  Centre  of  gravity  of  any  volume  generated  by  the  motion  of  a  varying 

surface  along  a  fixed  axis  pcri>endicular  to  its  plane,  and  passing 

through  its  centre  of  gravity             -             -             -             -             -  68 

47.  Expressions  for  the  co-ordinates  of  the  centre  of  gravity  of  a  surface  or 

solid  which  is  not  symmetrical          -             -             -             -             -  69 

48.  The  theorem  of  GtiUUn           -             -             -             -             -             -  70 

49.  On  the  equilibrium  of  a  solid  body  acted  on  by  any  system  of  forces 

whatever     -  -  -  -  -  -  -  -73 

50.  Determination  of  the  resultant  of  a  system  of  forces  situated  in  one 

plane,  and  applied  to  different  points  of  the  body         -             -             -  ib. 

51.  The  moment  of  the  resultant  of  this  system  of  forces  is  equal  to  the 

sum  of  the  moments  of  the  components         -             -             -             -  74 

52.  Equations  necessary  to  determine  the  resultant  in  intensity  and  direc- 

tion -  -  -  -  -  -  -  -75 

53.  Equations  of  equilibrium  of  a  system  of  forces  in  one  plane,  acting  on 

different  points  of  a  body     -             -             -             -             -             -  76 

.54.  Examination  of  the  import  of  the  several  equations  of  condition           -  77 

55.  Determination  of  the  equations  of  equilibrium  when  the  acting  forces 

are  situated  in  different  planes         ....  ib 

56.  Examination  of  the  import  of  the  several  equations  of  condition         -  79 


CONTENTS'  9 

Article  _  _  ■Pag'C 

57.  Conditions  of  equilibrium  when  there  is  an  immoveable  point  in  the 

body  or  system         -  -  -  -  -  -  -81 

—  Conditions  of  equilibrium  when  there  is  a  fixed  axis  in  the  system       -     ib. 

58.  Analogy  between  the  theory  of  moments  and  the  theory  of  projections 

in  geometry  -  -  -  -  -  -  -82 

59.  Problems  on  the  equilibrium  of  a  solid  body    -  -  -  -     84 

—  A  bent  lever  ACB  (fig.  47,)  is  suspended  at  C,  about  which  point  it  is 

free  to  move  in  a  vertical  plane,  and  weights  are  attached  to  its  ex- 
tremities :  to  find  the  position  in  which  it  will  rest    -  -  -     ib. 

60.  An  obhque  cylinder  stands  on  a  horizontal  plane  :  to  determine  the  great- 

est weight  that  will  hang  suspended  from  its  upper  edge  without  over- 
turning it-  -  -  -  -  -  -  -85 

61.  One  extremity  of  a  heavy  rod  is  moveable  about  a  fiLxed  point  in  a  ver- 

tical plane,  and  to  the  other  extremity  is  fastened  a  cord  which  goes 
over  a  pulley  in  the  same  horizontal  line  with  a  fixed  end,  and  sup- 
ports a  weight  equal  to  half  the  weight  of  the  rod  :  to  determine  the 
position  in  which  the  rod  will  rest     -  -  -  -  -    ib. 

62.  Two  heavy  bars,  moveable  each  in  the  same  vertical  plane  about  one 

extremity,  mutually  support  each  other  :  to  determine  their  position     86 

63.  A  beam  of  variable  thickness  has  one  end  suspended  by  a  cord  of  given 

lengtli,  fixed  at  a  given  point  above  an  incUned  plane  of  given  incli- 
nation, and  the  other  end  of  the  beam  is  sustained  by  the  inclined 
plane :  to  determine  the  position  of  the  beam,  weight  sustained  by 
the  cord,  and  pressure  against  the  plane,  when  the  beam  is  at  rest     -     87 

64.  A  given  beam  is  supported  by  strings  which  go  over  pulleys,  and  have 

given  weights  attached  to  them  :  to  find  the  position  of  equilibrium       88 

65.  A  given  beam  hangs,  by  two  strings  of  given  lengths,  from  two  fixed 

points  :  to  find  the  position  of  equilibrium  -  -  -     89 

66.  A  given  solid  hemisphere,  with  its  convex  part  upon  a  smooth  inclined 

plane  of  given  inclination,  is  kept  from  sliding  by  a  string  of  given 
length,  fastened  at  one  end  to  a  given  point,  and  at  the  other  to  the 
edge  of  the  hemisphere :  to  determine  the  point  where  the  hemisphere, 
when  at  rest,  touches  the  plane,  the  pressure  on  the  plane,  and  the 
tension  of  the  string  -  -  -  -  -  -     89 

67.  To  determine  the  position  in  which  a  paraboloid  will  rest  upon  a  hori- 

zontal plane  -  -  -  -  -  -  -90 

68.  To  determine  the  pressures  exerted  by  a  door  on  its  hinges     -  -    91 

69.  73.  Additional  problems  for  exercise  -  -  -  -     92 

74.  On  the  mechanical  powers  -  -  -  -  -  93 

75.  Theory  of  the  lever     -  -  -  -  -  -  -  ib. 

76.  Advantage  of  a  combination  of  levers  -  -  -  -  94 

77.  A  body  being  weighed  successively  in  the  two  scales  of  a  false  balance ; 

to  determine  the  true  weight  -  -  -  -  -     ib. 

78.  Construction  of  the  common  steelyard  -  -  -  -     ib. 

79.  To  determine  what  must  be  the  length  of  one  arm  of  a  lever  of  the 

second  kind,  in  order  that  a  weight  at  the  extremity  of  the  other  may 

be  supported  by  the  least  power  possible  -  -  -  -  95 

80.  Theory  of  the  wheel  and  axle              -  -  -  -  -  ib. 

81.  Of  toothed  wheels      -             -             -  -  .  -  -  97 

82.  On  the  figure  of  the  teeth  of  wheels     -  -  -  -  -  ib. 

83.  Theory  of  the  pulley  "             -             -  -  -  -  -  98 

84.  Advantage  of  a  system  of  pulleys        -  -  -  -  -  99 
85,86.  On  difierent  systems  of  pulleys    -  -  -  -  -  100 

2 


10  CONTENTS. 

Arliele  Pog' 

87.  Thcon' of  the  inclined  plane               -            -            -  -  -  100 

88.  Thfory  of  the  screw               -             -             -             -  -  -  101 

89.  On  die  wedge             -             -             -             -             -  •  -  103 

90.  Theory  of  the  wedge  abstracting  from  the  influence  of  friction  -  104 

91.  On  the  strength  and  stress  of  l)caras     -             -             -  -  -  106 

92.  Galileo's  hypothesis  of  the  resistance  of  the  fibres       -  -  -  ib. 

93.  The  hypothesis  of  Leibnitz                 ...  -  107 

—  Results  of  ,1/r.  Barloivs  experiments  -  -  -  -   1 08 

94.  Problems  on  the  strength  of  beams  in  various  positions  -  -     ib. 

95.  Useful  deiluctions  from  the  preceding  problems,  as  to  the  most  econo- 

mical forms  of  beams  -  -  -  -  -  -111 

96.  Method  of  determining  the  actual  weight  which  any  given  beam  is 

competent  to  sustain  in  given  circumstances  -  -  -     ib 

97.  Remarks  on  the  erroneous  notions  of  Emerson  and  others,  respecting 

the  comparative  strengths  of  beams  when  firmly  fixed  at  each  end, 
and  when  but  loosely  supported        -  -  -  -  -   114 

—  To  determine  the  strongest  rectangular  beam  that  can  be  cut  out  of  a 

given  cylindrical  tree  -  -  -  -  -  -115 


PART   II. 

ELEMENTS   OF   DYNAMICS. 

SECTION    1. 

On  the  Rectilinear  Motion  of  a  Free  Point. 

98.  Preliminary  remarks  -  -  -  -  -  -117 

99.  Of  inertia       -             -             -             -             -             -             -             -  ib. 

100.  Equations  of  uniform  motion               -             -             -             -             -  118 

101.  Expression  for  the  velocity  in  variable  motion              -             -             -  119 

102.  Expression  for  the  force  when  constant            -             -             -             -  ib. 

103.  Expression  for  the  force  when  variable            ....  120 

104.  Remarks  on  the  methods  of  representing  force             -             .             -  ib. 

105.  Explanation  of  the  true  meaning  of  the  foregoing  differential  expres- 

sions for  velocity  and  force  -  -  -  -  -121 

106.  Table  of  the  equations  of  uniform  and  accelerated  motion        -            -  122 

107.  On  the  force  of  gravity           ......  124 

108.  Examples  on  the  vertical  motion  of  heavy  bodies         -             -             -  125 

109.  On  the  motion  of  bodies  along  inclined  planes            ...  127 

110.  Examples  of  this  motion         .-..-.  128 

111.  Additional  problems                 ......  129 

112.  On  the  motion  of  projectiles  ......  131 

113.  Problems  on  projectiles  -  -  -  -  -  -132 

114.  Table  of  the  equations  containing  the  theory  of  projectiles       -             -  135 

115.  On  the  rectilinear  motion  produced  by  a  variable  force             -             -  136 

116.  To  determine  the  vertical  motion  of  a    heavy  body  towards  the  earth; 

the  force  of  gravity  varying  inversely  as  the  square  of  the  distance 

from  the  centre         -             -             .             .             .             -             -  ib. 

117.  To  determine  the  motion  of  a  body  attracted  towards  a  fixed  centre, 

the  force  varying  directly  as  the  distance       -             -             .             .  138 

118.  To  determine  the  motion  of  a  heavy  body  near  the  earth's  surface,  con- 

sidering the  resistance  of  the  air  to  vary  as  the  square  of  the  velocity  140 


CONTENTS.  11 

SECTION   II. 

On  the  Theory  of  Curvilinear  JMution. 
Article  Page 

119.  Preliminary  remarks  -  -  -  -  -  -143 

120.  Determination  of  the  general  equations  of  the  curvilinear  motion  of  a 

material  particle      -  -  -  -  -  -  -144 

121.  The  dilTerential  expression  for  the  velocity  in  the  curve  is  always  exact 

when  the  motion  is  due  to  any  number  of  central  forces       -  -   146 

122.  The  trilineal  spaces  described  by  the  radius  vector  about  a  single  centre 

of  force,  are  to  each  other  as  the  times  of  describing  them     -  -  148 

123.  Expression  for  the  tangential  force,  and  useful  deductions  from  it        -   149 

124.  Deduction  of  the  several  equations  of  motion  employed  in  last  section 

from  the  more  general  theory  established  in  this       -  -  -     ib. 

125.  On  the  motion  of  a  body  constrained  to  move  on  a  given  curve  -   150 

126.  General  expression  for  the  resistance  or  normal  force  at  any  point  of 

the  constraining  curve  ......   152 

—  To  determine  the  centrifugal  force  at  different  places  on  the  earth's 

surface,  from  knowing  the  time  of  one  rotation  on  its  axis  -  -  154 

127.  On  the  simple  pendulum  .  -  -  .  .   157 

128.  Formulas  concerned  in  the  theory  of  the  pendulum     -  -  -     ih. 

129.  To  determine  the  time  of  oscillation  in  a  circular  pendulum  -  -  158 

130.  To  determine  the  compression  or  ellipticity  of  the  earth  by  means  of 

seconds'  pendulums  -  -  -  -  -  -162 

131.  To  determine  the  time  of  oscillation  in  a  cycloidal  pendulum  -   163 

1 32.  When  a  body  vibrates  in  a  circular  arc,  to  determine  the  tension  of  the 

string  at  any  point  -  -  -  -  -  -164 

133.  To  determine  the  centrifugal  force  and  the  tension  of  the  string  in  the 

cycloidal  pendulum               -              -              -              -              -              -  165 

134.  To  determine  the  time  of  gyration  of  a  conical  pendulum        -             -  166 

135.  On  the  theory  of  central  forces            -             -             -             -             -  ib. 

136.  Expression  for  the  central  force  in  terms  of  the  perpendicular  on  the 

tangent       -  -  -  -  -  -  -  -  168 

137.  Expression  for  the  force  in  terms  of  the  chord  of  curvature      -  -  169 

138.  Expressions   for  the  centrifugal   force,  centripetal   force,    paracentric 

force,  &c.    -  -  -  -  -  -  -  -ib. 

139.  Motions  of  the  planets — Kepler's  laws  -  -  -  -  170 

140.  The  force  which  retains  the  planets  in  their  orbits  must  necessarily  be 

directed  towards  the  sun       --....  171 

141.  The  law  of  the  attractive  force  determined       -             -             -             -  ib. 

142.  Expression  for  the  velocity  at  any  point  of  the  orbit    -             .             -  172 

143.  Another  expression  for  the  velocity     -             -             -             -             -  173 

144.  The  attractive  force  like  terrestrial  gravity  influences  all  bodies  alike    -  ib. 

145.  If  a  body  move  in  a  conic  section  in  virtue  of  a  central  force,  that  force 

must  be  in  the  focus,  and  must  vary  inversely  as  the  square  of  the 
distance  at  which  it  acts       -  -  -  -  -  -174 

146.  Conversely  ;  if  the  central  force  vary   inversely  as  the  square  of  the 

distance,  the  body  must  describe  a  conic  section  having  that  force  in 

its  focus       -  -  -  -  ...  .  -ib. 

147.  Determination  of  the  particular  form  of  the  orbit  from  knowing  the  ini- 

tial velocity  of  the  planet  and  its  initial  distance  from  the  sun  -   175 

—  Expressions  for  the  intensities  of  the  solar  and  planetary  attractions    -  1 76 


12  CONTENTS. 

SECTION   III. 

On  the  Motion  of  a  Solid  Body. 

JirticU  Pogt 

148.  Preliminary  remarks               -             -             -    .         -             -  -   177 

149.  Expressions  for  the  moving  force,  velocity,  &c.  of  a  solid  body  -  178 
160.  On  the  collision  of  bodies       -             -             -             -             -  -179 

151.  Direct  impact  of  inelastic  bodies  .  .  .  -  .   180 

152.  Direct  impact  of  elastic  bodies  .....   \%\ 

153.  On  oblique  impact      -  -  -  -  -  -  -183 

154.  The  principle  of  D'Alembert  -  -  -  -  -  184 

155.  On  the  moments  of  inertia     ------  187 

156.  To  determine  the  moment  of  inertia  with  respect  to  an  axis  by  means 

of  the  known  moment  with  respect  to  some  other  axis  parallel  to  it  -   190 

157.  Rotation  of  a  solid  body,  acted  upon  by  impulse  about  a  I'lxcd  axis      -  192 

1 58.  Rotation  of  the  body  when  every  particle  of  it  is  actuated  by  an  accele- 

ralive  force  -  -  -  -  -  -  -193 

159.  Formulas  for  the  determination  of  the  centre  of  oscillation  of  a  vibra- 

ting body  on  copper             ......  jh. 

160.  Examples  of  the  determination  of  the  centre  of  oscillation        -             -  196 

161.  Determination  of  the  centre  of  percussion         ....  igg 

162.  On  the  centre  of  spontaneous  rotation              -             -             -             -  199 

163.  Determination   of  the   progressive  and  angular  velocities  of  a  body 

moving  in  consequence  of  an  impulsion  which  does  not  pass  through 
the  centre  of  gravity  ......  202 

1 64.  To  determine  the  distance  from  the  centre  of  gravity  at  which  the  im- 

pulsion must  have  been  given  to  produce  the  progressive  and  rotatory 
motion  observed  in  any  body  .....  204 

—    Application  of  this  to  the  double  motion  of  the  earth  -  -  -     ib. 

165.  A  variety  of  dynamical  problems  illustrative  of  the  preceding  theories  -  205 

166.  On  the  general  theory  of  the  motions  of  a  system  of  bodies  acted  on  by 

any  accelerative  forces  whatever       -  -  -  -  -215 

167.  Motion  of  the  system  when  the  only  forces  acting  are  the  mutual  at- 

tractions of  the  bodies  -  -  -  -  -  -217 

168.  The  general  principle  of  the  conservation  of  areas       -  .  .  218 

169.  The  general  principle  of  the  conservation  of  vis  viva  -  -  -     ib. 

170.  On  the  composition  of  rotatory  motions  ....  220 

171.  On  the  principal  axes  of  rotation         .....  223 
—     Through  every  point  of  space  three  principal  axes  may  always  be 

drawn         --.-....  224 

172.  The  same  shown  otherwise     .--...  226 

173.  If  a  body  revolve  about  a  principal  axis  through  the  centre  of  gra%'ity 

this  axis  will  suffer  no  pressure        .....  229 

174.  Further  particulars  respecting  the  rotation  about  a  principal  axis  -  230 

175.  Equations  of  the  rotatory  motion  of  a  body  about  its  centre  of  gravity 

when  fixed  -  -  -  -  -  -  -     ib. 

176.  If  the  instantaneous  axis  of  rotation  does  not  coincide  with  a  principal 

axis  at  the  commencement  of  motion  it  can  never  aftervyards  coincide 
with  one     -  .  .  ....  23' 

177.  Determination  of  the  circumstances  of  the  rotation  when  the  instanta- 

neous axis  is  nearly  coincident  with  a  principal  axis  at  the  com- 
mencement .......  233 

178.  A  collection  of  miscellaneous  dynamical  problems      ...  234 

179.  Note,  Poissoft'o  proof  of  the  parallelogram  of  forces    -  -  -  236 


PART  I. 

ELEMENTS  OF  STATICS. 
INTRODUCTION. 

Artide  (1.)  Mechanics,  taken  in  its  most  extensive  acceptation, 
is  the  science  which  embraces  all  inquiries  respecting  the  equilibrium 
and  motion  of  bodies,  whether  solid  or  fluid,  and,  therefore,  consti- 
tutes a  very  large  and  important  part  of  Natural  Philosophy,  or 
that  vast  body  of  knowledge  which  explains  the  laws  that  govern 
the  various  operations  of  nature.  It  is  usual  however  to  give  a  more 
limited  signification  to  the  term  Mechanics,  and  to  treat  under  that 
denomination  only  of  the  equilibrium  and  motion  of  solid  bodies. 
The  theory  of  the  equilibrium  of  solid  bodies  is  called  Statics  ; 
the  theory  of  their  motion  Dynamics  ;  these,  therefore,  are  the  two 
great  branches  of  the  science  of  Mechanics. 

(2.)  If  a  body  be  submitted  to  any  influence  which  would,  if  not 
opposed  by  an  equal  counteracting  influence,  move  it,  such  influ- 
ence, whatever  it  be,  is  called  force.  The  term  force,  therefore,  as 
employed  in  Mechanics,  applies  not  merely  to  what,  in  common 
language,  we  understand  by  power  or  physical  energy,  but  also  to 
every  cause  which  either  produces  or  tends  to  produce  motion, 
however  hidden  or  inexplicable  that  cause  may  be. 

(3.)  A  body  subjected  to  the  action  of  a  force  or  moving  influ- 
ence, ought,  necessarily,  to  move  in  the  direction  of  that  force ; 
towards  it,  if  the  force  draw  or  attract  it,  and  from  it,  if  the  force 
push  or  repel  it.  Hence,  it  is  the  tendency  of  a  body  influenced 
by  a  single  force  applied  to  it  at  rest  to  move  in  a  straight  direction. 
But  if  to  the  same  point  two  equal  and  directly  opposite  forces  are 
applied,  the  tendency  to  motion  in  one  direction  being  equal  to  the 
tendency  in  the  opposite  direction,  the  point  will  necessarily  remain 
at  rest,  and  will  be  as  much  prepared  to  obey  the  influence  of  any 
third  force  as  it  was  before  the  two  counterbalancing  forces,  of  which 
we  have  just  spoken,  were  applied. 

It  is  obvious,  that  although  two  equal  forces  be  applied  to  the 
same  point  they  will  not  keep  that  point  at  rest,  unless  they  are  di- 
rectly opposite  as  well  as  equal.  This  indeed  may  be  easily  proved 
thus  :  Suppose  two  equal  forces  P  and  Q  (fig.  1 ,)  tend  to  draw  the 
point  M  in  their  respective  directions  MP,  MQ,  which  are  not  oppo- 
site to  each  other  ;  if  we  suppose  the  point  to  remain  at  rest,  let  us 
B  13 


14  INTRODUCTION. 

introduce  a  third  force  P'  equal  and  opposite  to  P,  that  is,  tending 
to  draw  M  towards  P'  with  the  same  energy  as  P  tends  to  draw  it 
towards  P.  Now  tlie  point  being  acted  upon  by  three  forces,  of 
which  two,  viz.  P  and  P',  are  in  eiiuilibriuni,  the  point  will  tend  to 
move  in  the  direction  MQ  of  the  third  force  Q.  But,  by  hypothesis, 
the  two  forces  P,  Q  are  in  equilibrium  ;  hence  the  tendency  to  mo- 
tion must  be  in  the  direction  MP'  of  the  third,  w'hich  is  absurd. 

(4.)  As  it  is  the  business  of  Statics  to  investigate  the  laws  of 
forces  in  equilibrium,  it  will  readily  occur  to  the  student  that  one  of 
the  principal  probleius  of  this  brancli  of  Mechanics  is  to  determine, 
from  knowing  the  magnitudes  and  directions  of  forces  applied  to  a 
point,  what  must  be  the  magnitude  and  direction  of  that  counterba- 
lancing force,  which  would  j)revent  motion  ensuing. 

We  may  estimate  the  magnitudes  of  the  forces  of  which  we 
speak  by  means  of  weights  :  for  it  is  plain  that  whatever  influence 
at  P  (fig.  2,)  solicits  M,  M  may  yet  be  kept  in  its  place  by  the 
counteraction  of  some  weight  W  tending  to  draw  M  in  the  opposite 
direction  MP'.  If  for  the  force  P  were  substituted  another,  which, 
in  order  to  keep  M  unmoved,  would  render  it  necessary  to  double 
the  weight  W,  we  should  then  say  that  this  new  force  was  double 
the  former,  and,  in  like  manner,  the  ratio  of  any  two  forces  acting 
separately  at  P  would  be  determined  by  the  ratio  of  the  separate 
counteracting  weights  acting  in  the  directing  MP'.  We  see  that 
weight  is  a  very  suitable  representative  or  measure  of  force  ;  but  for 
all  the  purposes  of  comparison  it  matters  not  by  what  we  represent 
a  force,  taking  care  only  that  the  representing  quantities  shall  have 
the  same  ratio  to  one  another  as  the  forces  represented,  we  are, 
therefore,  at  liberty  to  choose  that  mode  of  representation  most  con- 
ducive to  the  ends  in  view,  viz.  the  investigation  of  the  theory  of 
equilibrating  forces.  We  accordingly  represent  a  force  by  a  line 
drawn  from  the  point  on  which  it  acts,  called  the  point  of  application, 
in  the  direction  of  the  force.  The  length  of  this  line  for  one  of  the 
forces  of  the  equilibrating  system  may  be  arbitrary,  but  for  any  other 
of  the  forces  the  length  of  the  representing  line  will  be  to  the  former 
as  the  represented  force  is  to  the  former  force.  Hence  the  theory 
of  statics  is  reduced  to  that  of  lines  and  angles. 

(5.)  We  have  just  said,  that  a  force  is  represented  by  a  line  drawn 
from  the  point  of  application  in  the  direction  of  that  force  ;  but  we 
are  at  liberty  to  consider  any  point  of  this  direction  as  the  point  of 
application  of  the  force,  and  not  merely  the  material  point  on  which  it 
acts  :  thus  it  matters  not  whether  the  force  acting  upon  M  (fig.  2,) 
to  pull  it  in  the  direction  MP  be  applied  to  the  point  P  or  to  any  other 
point  in  MP,  provided  we  consider  MP  to  be  a  perfectly  inextensible 
line  connecting  P  with  M.  Or  if  P  be  a  repulsive  force  tending  to 
push  M  in  the  direction  MP'  then,  supposing  MP  to  be  a  perfectly 


INTRODUCTION'.  15 

rigid  line,  it  matters  not  whereabouts  in  this  line  P  may  be.  All  this 
is  too  obvious  to  require  any  laboured  proof,  and  we  shall  now 
proceed  to  the  general  theory  of  equilibrating  forces  acting  on  a  free 
point,  viz.,  the  point  where  the  directions  of  these  forces  meet,  and, 
to  avoid  confusion,  we  shall  generally  consider  the  forces  concerned 
as  pulling  forces ;  for  a  pushing  force  may  obviously  be  always 
supplied  by  a  pulling  force  of  the  same  intensity,  and  acting  in  an  op- 
posite direction. 


JOHW  S.  PRELL 

\oil  &  Mechanical  Engineen 

8AN  FRANCISCO,  CAU 

SECTION  I. 

ON  THE  EQUILIBRIUM  OF  A  POINT. 


CHAPTER  I. 

ON  THE  COMPOSITION  AND  RESOLUTION  OF  CONCURRING  FORCES. 

Article  (6,)  By  concurring  forces  we  are  to  understand  forces 
of  which  the  directions  all  meet  in  a  point,  upon  which  point  they 
simultaneously  act;  and  they  are  said  to  be  situated  in  the  same 
plane  when  their  directions  are  all  in  the  same  plane.  To  deter- 
mine the  resultant  of  two  such  forces,  that  is,  a  single  force  equally 
effective  with  the  two,  is  a  problem  of  great  importance,  and  to  this 
the  present  chapter  will  be  chiefly  devoted.  The  simplest  case  is 
that  in  which  the  concurrent  forces  act  not  only  in  the  same  plane, 
but  even  in  the  same  straight  line ;  if  in  this  case  the  forces  should 
both  conspire  or  tend  to  move  the  point  in  the  same  direction,  then 
their  resultant  would  be  equal  to  their  sum ;  for  it  is  plain  that  the 
weight  W  (fig.  2,)  has  the  same  effect  on  M  as  two  smaller  weights, 
together  equal  to  W.  If  the  two  forces  are  opposite,  then  their  re- 
sultant must  be  equal  to  their  difference,  and  the  direction  of  it  to- 
wards the  greater  force,  because  so  much  of  the  greater  force  as  is 
equal  to  the  less  force  which  opposes  it,  is  employed  in  keeping 
the  point  in  equilibrium,  and  the  tendency  to  motion  is  the  effect  of 
the  remaining  force. 

If,  instead  of  two  conspiring  forces,  there  were  three,  then,  by 
adding  together  two,  we  should  obtain  the  resultant  of  those  two, 
and  this,  added  to  the  third,  would  furnish  the  resultant  of  the  three ; 
and  in  like  manner  the  resultant  of  four,  or  of  any  number  of  con- 
spiring forces,  is  found  by  merely  adding  together  the  component 
conspiring  forces.  If  there  be  two  systems  of  conspiring  forces 
directly  opposed  to  each  other,  they  may  thus  be  reduced  to  a  single 
pair  of  opposing  forces,  and  the  difference  of  these  two  will  be  the 
resultant  of  the  whole  system.  The  direction,  as  well  as  the  mag- 
nitude of  this  resultant,  will  be  expressed  algebraically,  if  we  agree 
to  consider  the  forces  which  conspire  in  one  direction  as  all  positive, 
and  those  which  conspire  in  the  opposite  direction,  as  all  negative  j 
for  we  may  then  say  that  the  resultant  of  any  number  of  forces  act- 
ing in  the  same  straight  line,  is  equal  to  the  sum  of  those  forces; 
b2  3  17 


vi<n^ 

18  ELEiMENTS   OF    STATICS. 

the  sign  of  tliis  sum  pointing  "out  the  direction.  This  tlieorem,  of 
course,  includes  the  case  in  which  the  forces  all  pull  one  way,  that 
is,  where  there  is  hut  a  single  system  of  conspiring  forces. 

Having  disposed  of  this  simple  case  of  the  general  problem,  we 
are  now  to  determine  the  magnitude  and  direction  of  the  resultant 
of  any  two  concurring  t'orces  situated  in  a  plane  ;  we  speak  of  only 
two  concurring  forces,  because,  as  we  shall  soon  see  here,  as  in  the 
case  just  considered,  that  when  we  know  how  to  compound  two, 
the  composition  of  any  number  can  present  no  difficulty. 

There  are  several  ways  of  arriving  at  the  solution  of  this  impor- 
tant problem,  but  the  simplest  process  with  which  we  are  acquainted 
is  that  given  by  Professor  Gregory,  in  his  valuable  treatise  on  Me- 
chanics, who  conducts  the  investigation  as  follows : 

(7.)  Prop.  The  equivalent  of  several  forces  situated  in  one  plane, 
is  in  the  same  plane. 

For  if  we  suppose  the  equivalent  to  be  out  of  the  plane  of  the 
forces,  on  either  side,  we  may  always  find  a  line  on  the  other  side 
of  the  plane  situated  in  a  perfectly  similar  manner  ;  and  since  there 
can  be  no  reason  why  the  resultant  should  he  in  one  of  these  direc- 
tions, rather  than  in  the  other;  it  is  therefore  in  neither  of  them,  un- 
less we  admit  the  absurd  consequence  that  it  is  in  both,  that  is,  unless 
we  admit  that  the  same  forces  acting  in  like  manner  can  produce  two 
distinct  eflects. 

Cor.  The  resultant  of  two  equal  forces  must  be  in  their  plane  ; 
and  it  must  be  in  the  line  which  bisects  the  angle  of  their  direction, 
since  there  is  no  reason  Avhy  it  should  tend  more  to  one  side  than  to 
another. 

(8.)  Prop.  If  to  a  material  point,  already  kept  in  equilibrio  by  a 
system  of  forces,  another  system  is  applied  also  in  equilibrio,  this 
will  not  destroy  the  pre-existing  equilibrium :  this  is  manifest. 

Cor.  Hence,  if  the  three  forces,  C,  C,  0,  (fig.  3,)  are  in  a  state 
of  equilibrium,  and  if  each  of  the  forces  were  doubled,  or  tripled, 
or  quadrupled,  &c.,  or  if  they  were  halved,  quartered.  Sic,  or  changed 
in  any  proportion,  the  equilibrium  would  remain  so  long  as  they  con- 
tinued to  act  in  the  same  directions  CP,  C'P,  OP. 

Cor.  2.  Hence,  also,  since  the  resultant  R  is  always  equal  and  op- 
posite to  one  of  the  forces  (as  O),  it  follows,  that  when  the  magni- 
tudes of  equilibriated  forces  concurring  in  a  point  are  made  to  vary 
in  any  ratio,  the  resultant  retains  its  position  but  changes  its  magni- 
tude in  the  same  ratio. 

(9.)  Prop.  If  three  equal  forces  are  inclined  to  one  another  in  an- 
gles each  of  120  degrees,  any  one  of  them  will  balance  the  joint  ac- 
tion of  the  other  two.  This  is  likewise  incontrovertible ;  for  neither 
of  the  forces  can  prevail. 

(10.)  Prop.  Two  equal  forces  inclined  in  an  angle  of  120  degrees, 


CONCURRIXG  FORCES  IN  ONE  PLANE.  19 

have  for  their  equivalent  a  third,  which  has  the  direction  and  pro- 
portion of  the  diagonal  of  the  rhombus  constructed  on  the  lines  which 
represent  the  forces. 

For  if  C,  C  are  the  forces  (fig.  3,)  acting  on  the  point  P,  the  force 
O  whose  measure  is  OP=CP=C'P,  and  is  situated  so  that  the  an- 
gles CPO  and  C'PO  are  each  equal  to  CPC,  will  (9)  ensure  the 
equilibrium.  But  RP,  the  measure  of  the  equivalent  R,  is  equal  and 
opposite  to  OP  (6);  therefore  CP=PR=C'P  and  because  angle 
CPR=60°=C'PR,CR=CP and C'R=C'P.  Consequently  CPC'R 
is  a  rhombus,  and  RP,  the  representative  of  the  equivalent  of  the 
forces  C,  C,  is  its  diagonal. 

Cor.  If  half  the  angle  CPC  be  denoted  by  a,  v/e  shall  have 
PD=PC  cos.  fl=C  cos.  a,  whence  the  equivalent  RP=2  C  cos.  a. 

(11.)  Prop.  Any  two  equal  forces  have  for  their  equivalent  the 
diagonal  of  the  rhombus  constructed  on  the  right  lines  which  repre- 
sent them  in  magnitude  and  direction. 

1.  If  this  proposition  be  true  with  regard  to  any  two  equal  forces 
C,  C  acting  in  the  directions  CP,  C'P  (fig.  4),  and  forming  Avith 
their  resultant  R  the  angles  CPR,  C'PR  each  for  example  equal  to 
a,  it  is  true,  likewise,  for  two  other  equal  forces  c,  c'  acting  accord- 
ing to  the  directions  c  P,  c'P,  which  bisect  those  angles.  In  this 
case  c  may  be  considered  (7  Cor.)  as  the  resultant  of  two  equal  forces, 
X  and  y,  acting  in  the  directions  CP,  RP  ;  and  in  like  manner,  c' 
may  be  considered  as  the  resultant  of  two  other  equal  forces,  x'  and 
?/',  acting  in  the  directions  C'P,  RP :  so  that,  in  lieu  of  the  two 
equal  forces  c,  c',  we  may  consider  four  equal  but  unknown  forces  x 
x',  y,  y'  acting  in  the  directions  just  assigned  them.  The  two  first 
of  these,  x,  x',  acting  in  the  directions  CP,  C'P,  have,  by  hypothe- 
sis, the  diagonal  of  the  rhombus  for  their  resultant:  that  is,  they 
are  equivalent  to  a  force  expressed  by  2  x  cos.  a  acting  in  RP, 
therefore  the  resultant  z  of  c  and  c'  will  be  equal  to  2y-\-2x  cos.  a; 
but  a?=y,  therefore  z=2x(l-t-  cos.  a).  Now  the  angles  CPR, 
c  P  c'  being  each  equal  to  half  CPC,  are  equal  to  each  other;  and 
c  being  the  resultant  of  two  equal  components  acting  in  CP  and 
RP,  we  have 

.  J^  '    "    '    ~  z 

Substituting  this  value  of  x  for  it  in  the  preceding  equation,  we  obtain 

2^2  

Z  = (l+COS.  «)  .-.  ;22_2c2(l  -fcos.  fl)  .'.  z  =  c  v/2(l+cos.  a). 

But  it  is  known  that  cos.  i  a=^        ^     — ,  (Gregory^ s    Trigono- 

7netry,  p.  46,)  whence,  by  substitution,  z=2c  cos.  5  a.     Conse- 
quently, the  proposition  if  true  for  a  is  true  for  I  a. 


20  ELEMENTS    OF    STATICS. 

2.  In  exactly  the  same  manner  may  the  proposition  be  proved 
true  with  respect  to  the  half  of  5  a  or  4  a,  and  in  succession  for 
i  a,  j\a,  ^^a,  6ic.  That  is,  since  it  is  true,  (9)  when  the  angle 
CPC  is  measured  by  5  of  the  circumference,  it  is  likewise  true 
■when  the  angle  between  the  equal  components  is  measured  by  ^, 
f  T'  IT'  ^^'  0*  *1'C  circumference,  where  the  series  may  be  continued 
sine  limite. 

3.  If  the  proposition  be  demonstrated  for  the  three  angles  a,  b, 
and  a  —  b,  it  will  be  true  for  the  angle  a 4-'^  »  that  is,  if  we  take  two 
equal  components  c  and  c',  making  with  their  resultant  x,  angles 
=a -\- b,  we  shall  have  a'=2c  cos.  (a-f6).  Thus,  if  in  fig.  5  the 
angles  CPR,  C'PR  are  each  equal  to  a,  and  cPC,  CPrf,  c'PC, 
C'Pd',  each  equal  to  b:  conceiving  two  forces  rfP,  d'P  each  equal 
to  c,  their  resultant  will,  by  hypothesis,  be  =2c  cos.  (a~-b),  because 
dPR=a  —  b ;  and  this  quantity  subtracted  from  the  resultant  of 
c,  c',  d,  d',  will  give  x.  But  c  and  d  have  their  resultant  C  act- 
ing in  CP,  and  =2c  cos.  b;  the  same  thing  holding  with  respect 
to  c'  and  d',  we  have  two  forces  equal  to  C  and  equivalent  to  one, 
which  is  2C  cos.  a  or  4c  cos.  a  cos.  b  ;  whence  a:=4c  cos.  o  cos. ft 

— 2c  cos.  (a—b).     But  cos.  a  cos.  b=i  cos.  (o+^y+d  cos. 
(a — b).     (See  Gregory's  Trig.,  p.  44,  art.  20).     Which  value  of 
cos.o  cos.  b,  substituted  for  it  in  the  preceding  equation,  gives  0'= 
2ccos.(o+  b).   So  that  the  proposition  when  true  for  a,  b,  and  a — b, 
is  true  for  a-\-b. 

4.  Let  b  be  taken  as  small  as  we  please  in  the  series,  1,  -j-V,  2^:j ,  -Jj- 
&c.  and  let  a  be  the  preceding  term  in  the  series,  then  o,  6,  a—b,a-\-b, 
are  26,  b,  b,  and  36,  respectively,  in  each  of  which  the  proposition 
holds:  again,  if  a=Sb,  a-^b=4b;  if  fl=46,  0+6  =  56,  &c.  So 
that  the  theorem  is  demonstrated  for  all  angles  in  the  series  6,  26,  36, 
46,  56,  &-C.,  in  which  6  may  be  taken  of  a  magnitude  less  than,any 
one  which  can  be  assigned.  Consequently,  the  theorem  is  true  with 
respect  to  any  rhombus  whatever  ;  for  let  any  rhombus  be  proposed, 
which  it  is  affirmed  is  an  exception  to  this  proof;  we  can,  it  is  ob- 
vious, by  choosing  6  lower  than  any  assigned  angle,  and  taking  a 
suitable  multiple  of  it,  approach  nearer  the  aecepted  angle  than  by 
any  assignable  difference,  that  is,  we  show  that  our  theorem  is  ap- 
plicable to  the  angle  itself.  j*  ., 

(12.)  Prop.  Any  two  forces  having  the  ratio  o^Hfe  sides  of  a 
rectangle,  and  whose  directions  coincide  with  those -^sMes,  hav^  for 
tlieir  equivalent  the  diagonal  of  that  rectangle. 

Let  the  two  forces  C,  C  (fig.  G,)  act  in  the  directions  CP,  CPS 
which  comprise   the   right   angle   P :   complete  the  parallelogram^ 
CPC'R,  and  draw  its  diagonals;  parallel  to  CC,  draw  cc'  termi- 
nated by  Cc,  C'c',  which  are  drawn  parallel  to  the  resulting  diagonal. 
Conceive  c  and  c'  to  be  two  equal  forces  acting  in  the  equal  lines  cP, 


CON'CURRING    FORCES    IN    ONE    PLANE.  21 

c'P,  opposite  to  each  other,  and  consequently  annihilating  each 
other's  effects;  then  cPDC  and  c'PDC  being  rhombi,  the  force  CP 
is  the  equivalent  of  cP,  DP,  and  C'P  that  of  c'P,  DP,  by  the  preced- 
ing proposition.  Therefore,  the  components  CP,  CP  are  the  same 
in  effect  as  the  opposite  ones  cP,  c'P  together  with  DP,  DP ;  that 
is,  the  equivalent  sought  is  2DP  or  RP,  the  diagonal  of  the  paralle- 
logram. 

Cor.  Since  RP  :  rad.  ::  CP  :  cos.  CPR  ::  C'P  :  cos.  C'PR,  we 
have  the  resultant  equal  to  either  component  divided  by  the  cosine 
of  the  angle  which  it  makes  with  the  resultant. 

(13.)  Prop.  Any  two  forces  whatever  have  their  equivalent  ex- 
pressed in  magnitude  and  direction  by  the  diagonal  RP  of  the  paral- 
lelogram constructed  on  the  lines  CP,  C'P,  which  represent  these 
forces. 

Having  completed  the  parallelogram  CPC'R  (fig.  7,)  on  the  given 
sides,  draw  cc'  perpendicular,  and  Cc,  C'c'  parallel,  to  the  diagonal ; 
demit  also  CD,  C'D' perpendicular  to  the  diagonal:  then  will  Cc 
PD,  C'c'  PD'  be  rectangles,  and  the  triangles  CRD,  C'PD'  equal  in 
all  respects,  consequently  Cc=DP,  RD=D'P,  and  cP=c'P.  The 
addition  of  the  equal  forces  c,  c' ,  acting  in  the  opposite  directions 
cP,  c'P,  will  make  no  difference  in  the  state  of  the  system ;  and 
since  the  components  DP,  cP,  have  CP  for  their  resultant,  and  the 
components  D'P,  c'P,  the  resultant  C'P,  (by  the  preceding  proposi- 
tion,) we  may,  instead  of  the  original  forces  CP,  C'P,  substitute  the 
forces  cP,  c'P,  DP,  D'P,  of  which  the  two  former  destroy  each 
other's  effects,  and  the  latter  DP,  D'P,  are  manifestly  equal  to  RP  ; 
that  is,  the  resultant  of  the  two  forces  CP,  C'P,  is  equal  to  the  dia- 
gonal RP  of  the  parallelogram. 

"  Thus  have  we,"  observes  Dr.  Gregory,  "  by  a  series  of  connect- 
ed* propositions,  demonstrated  that  which  is  justly  reckoned  the 
most  important  in  the  theory  of  Statics,  and  which  is  now  commonly 
spoken  of  under  the  title  of  the  Parallelogram  offerees.  The  de- 
monstration here  given  is  commenced  upon  the  same  principle  (9) 
as  that  proposed  by  D^Membert,  in  the  Memoirs  of  the  French  Aca- 
demy for  1769 :  it  was  somewhat  simplified  by  Francceur  in  his 
Mechanics  ;  but  what  is  here  offered,  at  the  same  time  that  it  is  more 
concise  than  the  demonstration  of  Francoeur,  is  freed,  it  is  hoped, 
from  some  objectionable  positions  into  which  that  author  has  certainly 
fallen." 

The  foregoing  proposition  of  the  parallelogram  of  forces,  has  also 
been  established  by  some  eminent  mathematicians  by  processes 
purely  analytical,  and  deduced  from  some  obvious  principle  neces- 
sarily involved  in  the  question  itself,  as  for  example,  that  if  two  equal 
forces  P,  P  concur  at  an  angle  e,  the  direction  of  the  resultant  must 
bisect  this  angle,  and  moreover  its  intensity  must  be  some  function 


22  ELEMENTS   OF    STATICS. 

of  P  and  5,  which  is  no  more  than  saying  that  the  intensity  of 
the  resultant  must  in  some  way  depend  on  the  intensity  of  its  equiva- 
lent components  and  on  their  mode  of  action.  This  is  the  condition 
from  wiiich  Poisson  sets  out  his  analytical  investigation.  We  have 
given  it  with  some  little  modification  at  the  end  of  the  volume. 

(14.)  From  what  has  now  been  proved,  it  follows  tliat  when  the 
intensities  and  directions  of  any  two  concurring  forces  are  given, 
the  determination  of  the  third  ibrce  equilibrating  these  will  be  re- 
duced to  the  determination  of  the  diagonal  of  a  parallelogram,  from 
liaving  the  two  sides  and  included  angle  given  ;  or  still  more  simply, 
it  will  be  reduced  to  the  determination  of  the  third  side  of  a  triangle, 
from  having  two  sides  and  the  included  angle  given  ;  for  if  PC,  PC 
are  the  two  given  forces  (tig.  7,)  then,  by  drawing  CR  equal  and 
parallel  to  PC',  the  diagonal  PR  will  be  just  as  well  determined  as 
if  we  had  constructed  the  parallelogram  CC,  so  that  instead  of  the 
sides  and  diagonal  of  a  parallelogram,  we  may  always  represent 
three  equilibrating  forces  as  well  in  direction  as  in  intensity  by  the 
three  sides  of  a  triangle  taken  in  order  as  PC,  CR,  RP;  so  that  as 
these  sides  are  as  the  sines  of  their  opposite  angles,  we  may  say 
that  the  intensity  of  any  one  of  three  equilibrating  forces  is  propor- 
tional to  the  sine  of  the  angle  between  the  directions  of  the  other 
two.  If  we  represent  the  sides  PC,  PC  by  P,  Q,  and  the  angle 
CPC  of  their  direction  by  a,  then,  since  in  the  triangle  CRP  the 
angle  C  is  the  supplement  of  a,  we  shall  have  this  analytical  ex- 
pression for  R,  viz.,  R3=P3-)-Q»-f  2PQcos.o,  so  that  the  funda- 
mental theorem  of  statics,  when  expressed  algebraically,  is  precisely 
that  which  is  also  the  fundamental  theorem  of  plane  trigonometry. 

(15.)  Knowing  how  to  compound  two  forces,  we  may  easily 
compound  several  or  determine  a  single  force  which  will  balance 
them,  and  that  either  by  geometrical  construction  or  by  analytical 
representation.  Thus,  suppose  four  forces  PC,,  PC^,  PC 3,  PC^, 
concurred  at  P,  then  we  miglit  proceed  geometrically  as  follows : 
Draw  in  the  plane  of  the  two  forces  PCj,  PC 2,  the  line  CjRj,  equal 
and  parallel  to  PC,,  then  PR,  will,  from  what  has  already  been 
proved,  be  the  equivalent  of  PC,,  PCg,  and  may  therefore  be  sub- 
stituted for  them  in  the  system.  Again,  in  the  plane  of  the  two 
forces  PR,,  PC 3,  draw  R,Rjj  equal  and  parallel  to  PC 3,  then,  as 
before,  PRj  will  be  the  equivalent  of  PR,,  PC 3,  that  is  of  the  three 
forces  PC,,  PC 3,  PC 3.  Lastly,  in  the  plane  of  the  two  forces  PR3, 
PC4,  draw  RgRg,  parallel  and  equal  to  PC^,  and  we  shall  then 
determine  PR3,  which  must  be  the  equivalent  of  the  whole  system. 
From  this  construction  it  is  obvious,  that  if  commencing  at  the  point 
of  concurrence  P  we  draw  successively  PC,,  CjR,,  RjRj*  Rgl^s' 
equal  and  parallel  to  the  several  forces,  then  the  line  PR3,  which 
closes  the  polygon  PC.RiR^RjP,  will  represent  the  resultant  of 


CONCURRING  FORCES  IN  ONE  PLANE.  23 

the  system,  and  this  in  whatever  planes  the  competent  forces  act, 
since  in  our  construction  we  have  not  confined  these  forces  to  any 
particular  planes.  Should  the  concurring  forces  be  in  equilibrium, 
the  last  of  the  points  Rj,  Rg,  R3,  &c.  will  fall  on  P,  making  the 
resultant  0. 

This  graphical  method  of  compounding  forces,  will  not,  however, 
answer  the  purposes  of  computation,  and  we  shall  therefore  now 
seek  a  general  analytical  expression  for  the  resultant  of  any  number 
of  concurring  forces,  confining  ourselves  first  to  those  only  which 
are  situated  in  one  plane. 

Determination  of  the  Resultant  of  any  Number  of  concurring 
Forces  situated  in  one  Plane. 

(16.)  We  have  already  seen  how  any  two  concurring  forces  may 
be  compounded  into  one,  but  before  we  can  conveniently  compound 
a  greater  number  we  must  first  reverse  this  process,  and  know  how 
to  resolve  any  single  force  into  two  concurring  forces  acting  in  cer 
tain  proposed  directions. 

Let  PR  (fig.  8,)  represent  any  given  force  acting  on  P,  and  let  it 
be  required  to  resolve  it  into  two  others  concurring  in  P,  and  acting 
in  the  directions  PC,  PC,  so  that  PC  may  make  a  given  angle  a 
with  PR,  and  PC'  may  make  a  given  angle  j3  with  PR.  Parallel  to 
PC  draw  RC,  then  (13)  making  PC  equal  to  RC,  PR  will  repre- 
sent the  force  of  which  PC,  PC  represent  the  components,  that  is, 
PC,  PC  Avill  be  the  components  sought.  Their  analytical  values 
will  be  obtained  by  determining  trigonometrically  the  two  sides  PC, 
CR  of  the  triangle  CPR,  from  having  the  side  PR  and  interjacent 
angles  given;  we  see,  therefore,  that  the  resolution  of  a  given  force 
in  any  two  arbitrary  directions  may  always  be  effected.  That  it 
may  be  effected  in  the  easiest  way  possible,  it  is  requisite  that  the 
proposed  directions  be  perpendicular  to  each  other,  for  then  the 
analytical  operations  are  expressed  simply  by  the  equations,  PC  = 
PRcos.a,PC'=PRcos.  )3,  where  a  and  j3  are  complements  of  each 
other.  When,  therefore,  we  have  to  decompose  a  force  into  two 
others,  and  are  at  liberty  to  choose  their  directions,  we  shall  always 
on  the  score  of  simplicity,  assume  these  directions  perpendicular  to 
each  other. 

(17.)  Let  it  now  be  required  to  compound  into  a  single  resultant 
the  several  concurring  P,  P^,  Pg,  P3,  &c.  (fig.  9). 

Assume  any  rectangular  axis  AX,  AY,  the  first  AX  making  with 
the  direction  of  the  several  forces  the  angles  a,  Oj,  a^,  aj,  &c.,  and 
the  second  AY  making  with  the  same  directions  the  angles  3,  /Sj, 
|3o,]83,  &c.  Then  by  resolving  each  of  the  proposed  forces  into 
two  others  acting  along  the  assumed  axes,  we  shall  have  for  the  sum 
X  of  all  the  components  kx,  Ax^,  AXj,  &c.  acting  along  kx, 


24  ELEMENTS    OF    STATICS. 

P  COS.  a  +  P,  COS.  Oj-f-Pj  COS.  Oj+Pj  COS.  03+<fec.  =  X  ....  (1), 

and  for  the  sum  Y  of  all  the  other  components,  that  is  those  acting 
along  AY 

Pcos. /3  +  P,  cos.  ^.-fPj  cos. /Sj-I-Pa  COS.  l3^-^iic.  =  Y  ....  (2), 

so  that  these  two  sums,  that  is  the  single  force  X  acting  along  AX 
and  the  single  force  Y  acting  along  AY,  may  be  substituted  for  the 
proposed  system  of  forces ;  hence  the  resultant  of  these  two  will  be 
the  resultant  of  the  original  system.  But  the  resultant  R  of  two 
forces  X  and  Y  acting  at  right  angles,  being  the  diagonal  of  the  rec- 
tangle XY,  is  R=v'  X^-f-Ys  .  .  .  .  (3),  this, therefore,  is  the  gene- 
ral expression  for  the  resultant  of  any  system  of  concurring  forces 
acting  in  a  plane. 

(18.)  It  might  at  first  sight  appear,  that  since  the  angles  /3,  p,, 
&c.,  are  the  complements  of  a,  o,,  (fee,  it  would  be  better  to  write 
in  the  first  member  of  (2)  the  expressions  sin.  o,  sin.  Oj,  &;c.  in- 
stead of  cos  ji,  cos.  i3,,  &;c. ;  such,  however,  is  not  generally  the 
case,  although  this  change  would  do  very  well  for  the  particular  ar- 
rangement of  the  forces  exhibited  in  the  figure,  for  it  is  easy  to  see 
that  this  arrangement  is  such  as  to  cause  all  the  components  acting 
along  each  axis  to  conspire.  If  one  of  the  forces  P^  were  directed  out 
of  the  angle  YAX,  as  in  fig.  10,  its  component  Ax^  would  obviously 
oppose  the  conspiring  forces  Ax,  Ax^,  Aa*,,  and  therefore,  agreea- 
bly to  (6),  its  analytical  representation  should  carry  a  contrary  sign, 
which  it  actually  does  do  when  we  give  it  the  form  P3  cos  a^,  but 
not  M'hen  written  P3  sin.  ^,  although  each  form  represents  the 
same  linear  magnitude  ;  similar  remarks  apply  when  P3  is  situated 
in  either  of  the  angles  XAY',  X'AY'.  Regarding  then  the  compo- 
nents which  act  in  the  directions  AX,  AY  as  positive,  and  those 
which  act  in  the  opposite  directions,  AX',  AY'  as  negative,  the  se- 
veral terms  in  (1)  and  (2)  will  have  their  proper  signs  involved  in 
those  of  their  cosines. 

(19.)  In  order  to  completely  determine  the  resultant  R,  we  must 
know  its  direction  as  well  as  its  intensity  (3).  Now  putting  a,  b, 
for  its  inclination  to  Ax,  AY,  we  know  that 

X  Y 

X=R  cos.  a,  Y=R  cos.  b  .'.  cos.  a=  -t^->  cos.  i=  -rj-, 

either  of  which  equations  makes  known  the  direction  of  the  resultant, 
so  that  the  resultant  will  be  completely  represented  by  the  equations 

R=^X2  +  Y2,cos.  a  =4-'?  •  •  •  •  (4). 
If  the  proposed  forces  are  themselves  in  equilibrium,  then  R=0,  so 
that  we  must  then  have  ^X^-^Y^—O  or  X^  +  Y^  =0 ;  but  as  every 


CONCURRING  FORCES  IN  ONE  PLANE.  25 

square  is  essentially  positive,  the  sum  of  two  cannot  be  0  unless 
each  separately  is  0,  so  that  when  the  forces  are  in  equilibrium,  we 
must  have  X=0,  Y=0,  showing  that  each  system  of  components 
must  be  in  equilibrium,  and  this  is  obviously  true  whatever  be  the 
inclination  of  the  axes  of  the  components. 

(20.)  As  a  particular  example  of  the  preceding  general  theory, 
let  us  suppose  four  forces,  P,  P,,  Pj,  P3,  concurring  in  a  point 
A,  of  which  the  intensities  are  respectively  denoted  by  the  num- 
bers 2,  3,  4,  5,  and   let  the  angles  included  by  their  directions 
be  PAP, =30°,     P,AP2  =  15°,      P^kP^=75°.     First  assume,  as 
above  directed,  two  rectangular  axes  AX,  AY,  and,  as  their  position 
is  arbitrary,  let  us  for  greater  simplicity  suppose  one  of  them  AX 
to  coincide  with  AP ;  then  the  inclinations  of  the  several  forces  to 
the  assumed  axes  will  obviously  be  as  follows : 
PAX=     0°=a  .-.  cos.  a  =1 
P,AX=  30°=ai      COS.  ttl=i^/3 
P2AX=  45°=a,      COS.  o,=i^/2 
P3AX  =  120°=a3        COS.  a3=—hy/B 


PAY=90°=/3  .-.  COS.  /3  =0 
PiAY=60°=/3,      COS.  ^,=1 
P2AY=45°=i3^       COS. /3,  =  i^2 
P3AY=30°=|33      COS.  133=1  v/3 
consequently,  the  two  general  equations 

P  COS.  a+P^  COS.  a,  +P2  cos.  a2  +&C.  =  X 
P  COS.  j3-i-Pi  COS.  |3i  -I-P2  COS.  /32  -}-&c.= Y 
become  in  this  case 

2+3^3+2^/2  — #y3=^X 
0+f+2v'2  +  fV3=Y; 
hence,  the  numerical  values  of  X  and  Y  being  thus  determined,  the 

value  of  ^=y/X2-i-y2  is  known,  and  thence  also  of 

X  Y 

cos.  a  =  -TT-j  or  cos.  b  =  -r^-. 

Determination  of  the  Resultant  of  any  Number  of  concurring 
Forces  situated  in  different  Planes. 

(21.)  It  was  necessary  before  we  could  compound  together  seve- 
ral forces  acting  in  one  plane,  first  to  determine  the  resultant  of  two, 
or  to  establish  the  parallelogram  of  forces,  so  likewise  before  we 
can  treat  the  more  general  case,  or  compound  together  forces  acting 
in  different  planes,  we  must  first  know  how  to  determine  the  result- 
ant of  three.  This  however  is  a  very  easy  matter,  it  has  indeed 
C  4 


26  ELEMENTS  OF  STATICS. 

been  aecomplisheil  jToomelrically  already  (15),  and  not  only  for 
three  but  for  any  number  of  forces.  But  let  there  be  three  P,  P,, 
P2,  concurring  in  A,  (Hit.  11,)  and  represented  by  the  lines  AB, 
AC,  AD;  then  we  know  from  the  article  just  referred  to,  that  it 
we  draw  BE  in  the  same  plane  with,  and  equal  and  jiarallel  to,  AC, 
and  then  EF  in  the  same  plane  with,  and  equal  and  parallel  to,  AD, 
the  line  AF,  which  closes  the  twisted  quadrilateral  ABEFA,  will 
represent  the  resultant.  Now  the  three  lines  AB,  BE,  EF,  are  ob- 
viously the  three  edges  of  a  parallelopiped  BG,  of  which  AF  is  the 
diagonal ;  hence  the  lines  representing  three  concurring  forces  not 
in  the  same  plane  form  the  edges  of  a  parallelopiped  whose  diago- 
nal is  their  resultant.  It  is  manifest  that  if  any  force  AF  be  pro- 
posed, and  we  draw  from  A  three  lines  AB,  AC,  AD,  in  any  direc- 
tions whatever,  not  all  in  the  same  plane,  nor  yet  any  two  in  the 
same  plane  as  AF,  we  may  construct  a  parallelopiped  having  these 
three  lines  for  edges,  and  AF  for  its  diagonal,  the  opposite  edges 
meeting  in  F.  Now  the  forces  represented  by  the  edges  of  this 
parallelopiped  meeting  in  A,  have  the  given  force  AF  for  their  re- 
sultant, therefore  this  given  force  has  these  three  for  its  compo- 
nents, so  that  any  force  may  be  decomposed  into  three  conctir- 
ri)ig  forces  acting  in  any  proposed  directions,  provided  all  three 
are  not  in  07ie  plane  nor  any  two  in  the  same  plane  as  the  proposed 
force. 

This  decomposition  will  be  most  easily  effected,  analytically, 
when  the  directions  of  the  components  are  at  right  angles ;  for  if 
a,  i5,  and  y,  represent  the  inclinations  of  the  proposed  force  R  to 
these  several  rectangular  directions,  then  the  three  components  will 
be  obviously  expressed  by  R  cos.  o,  R  cos.  ji,  R  cos.  y  .  (1),  for  in 
fact,  these  components  are  no  other  than  the  projections  of  the  ori- 
ginal force  on  three  rectangular  axes,  calling  these  several  projec- 
tions or  component  forces,  X,  Y,  Z,  we  have,  by  adding  their 
squares,  X2+Y2-fZ2=R3  (cos.2  a+cos.a  |3  +  cos.2  y), 
but  (.^nal.  Geom.  p.  228,  cos.^  a+cos.a  j3+cos.''  y=l  .  .  (2), 
.-.  R=v/X"2TV=  +  Z2 

It  thus  appears  that  when  we  wish  to  decompose  a  given  force  R 
in  three  rectangular  directions  AX,  AY,  AZ,  making  with  R  any 
proposed  angles  a,  /3,  y,  we  shall  have,  for  the  analytical  values  of 
the  components,  the  expressions 

X=Rcos.a,  Y=Rcos.(3,  Z=Rcos.y  ....  (3), 
and  when  on  the  other  hand  we  wish  to  compound  three  given  rec- 
tangular forces  X,  Y,  Z,  the  intensity  of  the  resultant  will  be  given 
by  the  expression  R=v/X2+Y=+Z*  .  .  .  .  (4), 
and  its  direction  by  the  expressions 

X  Y  '    Z 

COS.  a  =  -^,  C0S.3=-T7-,  COS.yrs-rg      i*^.      (5). 

K  K  K       '  ' 


CONCURRING    FORCES    IN    GENERAL.  27 

Two  of  these  latter  equations  are  however  sufTicient  to  fix  the 
position  of  the  resultant ;  since,  on  account  of  the  necessary  condi- 
tion (2),  any  one  of  the  cosines  become  fixed  when  the  other  two 
are ;  thus,  when  a  and  /3  are  determined,  we  get  y  by  the  equation 

COS.  y=v/l cos. 2  a COS.  2^  ....  (6). 

We  need  not  embarrass  ourselves  here  with  any  inquiry  about  the 
ambiguity  of  the  sines  of  the  cosines  in  (5),  arising  from  the  ambi- 
guity of  the  sine  of  the  radical  R  in  (4)  ;  for,  as  we  know  that  the 
resultant  must  necessarily  lie  within  the  angle  formed  by  X,  Y,  Z, 
the  angles  a,  /3,  y,  which  it  forms  with  these  lines,  must  always  be 
acute. 

(22.)  Let  us  now  proceed  to  determine  the  resultant  of  any  num- 
ber of  concurring  forces  P,  Pj,  Pg,  P3,  &c.,  situated  in  space,  and 
acting  in  given  directions. 

Through  the  point  of  concourse  A  draw  three  rectangular  axes 
AX,  AY,  AZ,  and  let  us  call 

o ,  ,3  ,  y ,  the  angles  which  P  makes  with  these  axes, 
«i'/5,,yi,       ....       P, 

»2'l^2'y2'         •        •        .        •         P2 
&C.  &C. 

then,  by  decomposing  each  force  according  to  these  axes,  Ave  have 
P  COS.  a  ,  P  cos.  i3  ,  P  cos.  y  for  the  components  of  P 

PjCOS.  a^,    PjCOS. /3i,    PjCOS.  y, P^ 

PgCOS.aa,    P2COS./32,    PsCOS.ya P^ 

&c.  &c. 

Adding  together  all  the  forces  which  act  in  each  axis,  the  three  sums 
X,  Y,  Z,  will  represent  three  rectangular  forces  acting  in  given  di- 
rections, which  may  be  substituted  for  the  proposed  system,  the 
values  of  these  three  forces  being 

P  COS.a  +  Pj  COS.  ttj-f  1*2  ^^^-  a2+*^^'  =  ^^ 

P  cos./3-i-PiCos.  fSj+Pj  cos./32  +  &c.  =  Y  I  .  .  .  .  (1). 

P  cos.  y-fPj  C0S.yj+P2   COS.  y2+&C.  =  Zj 

Having  thus  reduced  the  system  of  forces  to  three,  we  have,  for  the 

intensity  of  the  resultant,  the  expression  R=v'X'^  +  Y^-j-Z2  .  (2), 

and  for  the  angles  a,  b,  c,  which  it  makes  with  the  axes,  the  expres- 

X  ,       Y  Z  ,^, 

sions  cos.  a=— — ,  cos.  6=— 5-,  cos.c=— =3-  ....  (.3). 
K  K  K 

In  this  way,  therefore,  we  may  completely  determine  the  resultant 
of  any  system  of  forces  situated  in  space,  when  we  know  the  inten- 
sity of  each  force,  and  the  angles  its  direction  makes  with  three 
rectangidar  axes.  The  cosines  in  (1)  necessarily  carry  with  them 
the  proper  sines  as  in  art.  (18). 

When  the  system  of  concurring  forces  is  in  equilibrium,  then, 
since  R=0.  we  must  have  X=*+Y^+Z^=0  ;  but  as  every  square  is 


28  ELEMENTS    OF    STATICS. 


essentially  positive,  this  cannot  be  unless  X=0,  Y=0,  Z=0,  that 
is  to  say,  we  must  luive  (aqua.  1,) 

P  cos.o+Pj  cos.  aj  4-^2  <^os.  02+&c.=01 
P  cos.>3-f-Pi  cos.,3,-fP2  cos.  f32+«fcc.=0  I  .  .  .  .  (4). 
P  cos.  y+Pj  cos.yi+P2  cos.  7'2+<fec.=0  J 
These,  therefore,  are  the  equations  of  equUibrium  of  a  system  of 
concurrin?  forces  P,  Pj,  Pj,  &c  situated  any  how  in  space. 

(23.)  The  projection  of  any  line  in  space  on  two  rectangular 
axes,  may  obviously  be  found  by  first  projecting  the  line  on  the 
plane  of  those  axes,  and  then  projecting  this  projection  on  the  axes 
themselves.  Hence,  if  a  system  of  forces  in  equilibrium  be  project- 
ed on  the  plane  of  xy,  the  forces  represented  by  these  projections 
will  be  also  in  equilibrium,  seeing  that  the  components  of  these 
forces  will  be  X=0  and  Y=0.  Now  the  position  of  one  of  the 
rectangular  planes,  as  the  plane  of  xy,  is  always  arbitrary ;  on 
this  account  therefore,  and  on  account  of  the  equations  (4)  it  follows, 
that  ivhen  any  number  offerees  arc  in  equilibrio,  their  projections 
upon  any  plane  or  ttpon  any  line  will  also  be  in  equilibrio. 

(24.)  As  the  resultant  R  or  AF  (fig.  11,)  of  any  system  of  forces 
is  the  diagonal  of  the  parallelopiped  whose  edges  are  the  components 
X,  Y,  Z,  or  AB,  AC,  AD,  it  follows  that  X,  Y,  Z,  are  no  other 
than  the  co-ordinates  of  the  point  F,  (Anal.  Geom.  p.  222.)  Know- 
ing, therefore,  the  co-ordinates  of  a  point  F  in  the  line  AF  passing 
through  the  origin  A,  we  know  enough  to  enable  us  to  write  the 
equation  of  this  line  or  of  the  resultant.    These  equations  are  (Anal. 

7a  Z 

Geom.p.226-7,)z=-r^a',  z=^^y.     If  we  remove  the  origin  of 

xL  jL 

the  rectangular  axes  without  altering  the  directions  of  the  axes,  so 
that  the  co-ordinates  of  the  point  of  concourse  A  may  be  x',  y',  z\ 

2 
then  the  equation  of  the  resultant  will  be  z — z'=-— -  (x — x' )■, 

A. 

2 

z—z' =-r^(y — y'),  or  the  line  will  be  equally  represented  by 

combining  either  of  these  equations  with  that  of  the  third  projection, 
VIZ.  x—x  =^(y—y  ). 

Having  now  established  the  general  theory  of  the  equilibrium  of  a 
free  point  acted  upon  by  any  number  of  forces  any  how  situated, 
we  shall  devote  a  short  chapter  to  the  application  of  this  theory  to 
particular  examples. 


PROBLEMS  ON  CONCURRING  FORCES.  29 

CHAPTER  II. 

PROBLEMS    ILLUSTRATIVE    OF    THE    PRECEDING    THEORY. 

(24.)  We  have  already  adverted  (4)  to  the  propriety  of  represent- 
ing balanced  forces  by  means  of  weights,  from  tlie  circumstance  that 
any  influence  tending  to  move  a  free  point  may  be  always  counter- 
acted and  rendered  nugatory  by  the  opposing  influence  of  some 
weight  communicating  to  the  point,  by  means  of  a  cord  attached  to 
both.  All,  therefore,  that  has  been  established  in  the  preceding 
chapter  respecting  tlie  equilibration  of  forces  acting  on  a  point,  ap- 
plies when  these  forces  are  weights  acting  on  a  point  through  the 
intervention  of  cords,  provided  only  that  we  consider  these  cords  to 
be  themselves  without  weight  or  thickness,  to  be  inextensible,  and 
to  be  perfectly  capable  of  moving  over  the  fixed  points  or  puUies,  (as 
at  fig.  13,)  employed  to  direct  the  influence  of  the  weights,  with 
perfect  freedom.  The  consideration  of  a  system  of  weights  thus 
acting  on  a  free  point  through  the  intervention  of  cords  and  fixed 
points,  introduces  the  consideration  of  two  new  modifications  of 
force,  viz.  Tension  and  Pressure. 

By  the  tension  of  a  cord  is  to  be  understood  its  tendency  to 
stretch  under  the  influence  of  an  appended  weight ;  as  this  tendency 
varies  with  the  weight,  this  is  taken  as  its  measure ;  so  that  when 
we  speak  of  the  tension  of  a  cord  as  a  force,  we  always  mean  the 
weight  which  produces  that  tension.  The  pressure  on  a  fixed  point 
is  measured  by,  or  is  equal  to,  that  force  which  must  be  applied  to 
it,  when  supposed  free,  to  keep  it  in  equilibrium. 

Problem  I— A  cord  PACBP,  passes  over  two  fixed  points  or  small 
pulleys  A,  B  in  the  horizontal  line  AB,  and  two  given  equal  weights, 
suspended  at  the  extremities  P,  P,  support  a  third  given  weight  W. 
It  is  required  to  determine  the  position  of  C,  (fig.  12). 

The  point  C  is  kept  in  equilibrium  by  the  equal  tensions  (P  or  P , ), 
of  the  strings  CA,  CB,  and  by  the  weight  W  acting  vertically. 
Hence,  by  resolving  the  forces  in  the  directions  of  two  axes  CX, 
C  Y,  the  one  parallel  and  the  other  perpendicular  to  AB,  the  forces 
in  each  axis  must  destroy  each  other.  Hence,  taking  first  the  com- 
ponents in  CX,  we  have  the  condition 

P  COS.  a  +  Pj  COS.  a,  =0  or  P  cos.  a — P,  cos.  a  =  0, 
from  which  we  immediately  infer  that  as  P=Pi  ,  a=a',  and  there- 
fore  /3=i3,,  so   that  the   triangles  ACY,  BCY  are  equal,  and  CY 
bisects  AB.     Again,  because  ]3=/3j,  the  components  in  CY  are 
2P  cos. /3,  and  Win  the  opposite  direction,  therefore  the  second  condi- 
tion is 

c2 


30  ELEMENTS  OF  STATICS. 

w 

2P  COS.  |3  —  "\V=0  .-.  COS. /3  =  sin.  B=  -rr^. 

This  equation  is  sufficient  to  determine  the  point  C  or  the  line  EC  ; 
but,  to  avoid  any  trigonometrical  computation,  let  us  put  for  cos.  ^ 

YC             BC         v/BV^+YC2        2P 
Its  equal  ^,  then    yu  = YC =  "W 


YC3  W2  ^4p2 \V3 

The  solution  of  this  problem  may  be  conducted  differently  as  fol- 
lows :  Parallel  to  CB  draAv  AE,  and  produce  the  perpendicular  CY 
till  it  meets  in  E  ;  then  the  triangle  ACE  thus  constituted  will  have 
its  three  sides  in  the  directions  of  the  three  balancing  forces,  and 
will,  therefore,  be  proportional  to  them  ;  hence,  as  the  tensions  of 
CA,  CB  are  equal,  the  sides  CA,  AE  are  equal,  so  that  the  perpen- 
dicular AY  bisects  CE  ;  also 

op  A  P 

P  :  W  : :  AC  :  EC=2YC  .-.      -^=^, 

W       YC 

and  the  remainder  of  the  solution  may  be  as  above. 

If  we  suppose  W=0,  then  tlie  above  expression  for  YC  shows 

that  YC=0,  as  it  obviously  ought  to  be;  that  is,  the  cord  will  be 

brought  into  a  horizontal  line  AB.     iJut  upon  no  other  hypothesis 

will  tliis  be  the  case,  except,  indeed,  we  suppose  the  weights  P  to 

be  infinitely  great,  for  however  small  W  be  assumed,  yet  so  long  as 

P  is  of  finite  magnitude  YC  will  have  a  finite  value,  and  can  never 

be  accurately  0,  so  that  it  is  impossible  for  any  two  weights  P,  Pj, 

however  great,  acting  as  in  the  figure,  to  draw  a  third  weight  W  ever 

so  small  up  to  the  horizontal  line  AB.     The  same  is  true  if  instead 

of  a  small  weight  W  attached  to  a  cord  without  weight,  we  consider 

the  cord  itself  to  have  weight :  we  may  therefore  say  with  Professor 

Whewell 

"  Hence  no  force,  however  great, 
Can  stretch  a  cord,  however  fine, 
Into  a  horizontal  line 
Which  shall  be  accurately  straight." 

As  W  increases  from  0,  YC  increases  and  becomes  infinite  when 
W=2P,  so  that  when  the  weight  W  is  cither  equal  to  or  greater 
than  2P,  there  can  be  no  equilibrium,  for  W  will  continually  descend 
drawing  up  the  weights  P,  P^. 

Problem  II. — Suppose  the  weights  P,  P,,  are  unequal,  and  that 
the  line  AB,  instead  of  being  horizontal,  makes  a  given  angle  BAB' 
with  the  horizontal  line  AB' :  to  determine  the  position  of  C, 
(fig.  13.) 


PROBLEMS  ON  CONCURRING  FORCES.  31 

The  solution  will  perhaps  be  most  easily  obtained  without  re- 
solving the  forces;  thus,  draw  AE  parallel  to  CB,  meeting  the  verti- 
cal line  CE  in  E,  then  the  three  sides  CA,  AE,  EC,  being  in  the 
directions  of  the  three  equilibrating  forces  P  j ,  P,W,  are  proportional 
to  them ;  hence,  knowing  the  proportion  of  tlie  three  sides  of  the 
triangle  AEC,  we  may  determine  its  angles :  we  may,  therefore, 
consider  the  angles  ACE  and  AEC=ECB  as  found;  consequently 
the  angle  CAD,  the  complement  of  ACE,  becomes  known,  and  DAB 
being  given  the  angle  CAB  becomes  known ;  hence,  in  the  triangle 
CAB,  we  have  the  side  AB  and  the  angles  C  and  A  to  determine  the 
two  sides  AC,  BC. 

Thus,  suppose  P=4t^,  Pi,=3tfe,  and  W=5j^  :  also  AB=6  feet, 
and  the  angle  DAB=30°,  then  the  angles  of  a  triangle  whose  sides 
are  AC=3,  AE=4,  EC=5,  are  CAE=90°=BCA,  AEC=36°  . 
54'=ECB;  hence  DCA=90°— ECB  =  53"  .  6',  and  consequently 
ABC=DCA— 30°=23°  .  6'.  We  thus  have  AB=6  feet,  ACB  = 
90%  ABC  =23°  .  6' ,  whence  AC =2  •  354,  and  BC=5  •  518  feet. 

Problem  III. — Two  equal  weights  P,  P^,  balance  themselves  over 
any  number  of  fixed  pulleys :  to  determine  the  pressure  on  each, 
(fig.  14.) 

Each  of  the  points  A,  B,  C,  &c.  are  kept  in  equilibrium  by  three 
forces,  viz.  by  the  equal  tensions  on  each  side  of  it  and  by  the  pres- 
sure it  sustains,  which  latter  is  therefore  equal  and  opposite  to  the 
resultant  of  the  two  equal  tensions  P.  Hence,  calling  the  angles  at 
A,  B,  C,  &c.  a,  b,  c,  &c.  we  have 

the  pressure  on  A=2P  cos.ia 
B=2Pcos.|6 
C=2Pcos.ic. 
&c.  &c. 

Problem  IV. — A  cord  ACB  of  given  length  is  fastened  to  two 
hooks  A  and  B,  and  a  weight  W  is  at  liberty  to  slide  by  means  of  the 
ring  C  freely  upon  this  cord  ;  at  what  point  will  it  rest?  (fig.  15.) 

It  is  obvious  that  if  the  two  hooks  were  in  a  horizontal  line,  as 
A',  B  the  weight  W  ought  to  settle  itself  at  a  point  symmetrically 
situated  with  respect  to  the  two  points  A',  B,  that  is  CA'B  will  be 
an  isosceles  triangle,  and  therefore  the  angles  A'CH,  BCF,  made 
with  the  horizontal  line  HF,  will  be  equal.  But  if  the  hook  be  at  A 
instead  of  at  A',  then  the  force  in  CA  being  the  tension  of  CA,  or  the 
pressure  upon  the  hook  A,  we  may  consider  the  hook  to  be  removed 
and  a  force  equivalent  to  this  pressure  to  be  applied  at  A;  but  it 
matters  not  at  what  point  of  its  direction  a  force  is  applied,  so  that 
C  will  continue  undisturbed  if  the  force  be  applied  at  A',  that  is,  it 
will  make  no  difference  as  to  the  position  of  C,  whether  the  hook  be 


32  ELEMENTS  OF  STATICS. 

at  A  or  at  A',  the  tension  of  CA  being  the  same  throug-hout;  hence 
the  angles  ACII,  BCF  are  equal,  and  the  tension  of  CA  equal  to 
that  of  CB.  Produce  AC  to  meet  the  vertical  ED  in  D,  then  the 
sides  of  the  triangle  BCD,  being  in  the  direction  of  the  forces,  are 
proportional  to  them  ;  this  triangle  is  moreover  isosceles  having  CB 
=  CD  on  account  of  the  equal  tensions  of  CB,  CD,  or  of  the  equal 
angles  BCF,  ACH  ;  hence  AD  is  equal  to  the  length  of  the  string. 
We  thus  have  given 

AB=a,  AD=/,  EAB=a  .-.  AE=a  cos,  a.,  EB=a  sin.  o 

„^ „„     v^ /'■'  —  o^'cos.'o  —  a  sin.o 

ED=^//2  _  a^  cos.»  a  ••.  BF=^^ ^ ; 

also,  since,  DE  :  EA  : :  BF :  FC 

a  cos. a  «Rin.,i 

.*.rC= — - —     1- 


2        (  >//2  o^cos.^o' 

these  values  of  BF,  FC  determine  the  point  C, 

As  to  the  pressure  p  on  the  hook  B  or  A,  we  have 

BF  :  BC  : :  nV  :  J3,  but  BF :  BC  :  :  ED  :  DA 

DA    ,,.                    / 
.   », VV= w 

•  -P-  2ED  2^//"^^=^^  cos.^tt 

Or  the  pressure  may  be  found  by  first  determining  the  angle  D  by 
means  of  the  sides  AE,  AD;  this  angle  being  equal  to  BCY  or  AC  Y, 
we  have,  by  calling  it  /3,  and  resolving  the  equal  tensions  p  along 

W 

the  axis  CY,  2d  cos.  i3=W.-.  n= 

-^  ^2  cos.  ^ 

In  the  very  same  way  the  problem  may  be  solved,  when,  instead 
of  a  weight  W  acting  vertically,  any  power  acting  obliquely  be  at- 
tached to  the  ring,  for  the  lines  BA',  AE  being  drawn  perpendicu- 
lar to  the  direction  of  this  force  and  ED  parallel  to  it,  the  above  rea- 
soning becomes  then  obviously  applicable  to  this  case. 

The  question  may  be  viewed  in  rather  a  difl'erent  manner  from 
that  above,  by  considering  that  as  the  cord  ACB  is  of  constant 
length,  the  point  C,  before  arrivmg  at  a  state  of  rest,  must  describe 
the  arc  of  an  ellipse,  and,  moreover,  that  the  place  of  rest  must  be 
at  the  lowest  point  possible;  hence  the  horizontal  line  HF  must  be 
a  tangent  at  C  to  the  ellipse  whose  foci  are  A  and  B,  seeing  that 
every  other  point  in  this  ellipse  is  above  that  line ;  hence,  by  the 
property  of  the  ellipse,  the  angles  BCF,  ACH  are  equal,  and,  con- 
sequently, the  tensions  are  equal,  because  their  components  in  the 
directions  CF,  CH  must  be  equal. 

Itis  very  clear,  although  we  have  not  assumed  it  above,  that  because 
the  cord  passes  freely  through  the  ring,  the  tension  of  the  part  CB 
must  be  communicated  to  the  part  CA,  for  nothing  hinders  this  com- 
munication, so  that  the  cord  will  be  equally  tense  throughout. 


CONCURRING  FORCES  IN  ONE  PLANE.  33 

Problem  V. — A  cord  of  given  length  passes  over  two  pullies, 
and  one  of  its  extremities  P  is  put  througli  a  small  ring  or  noose  at 
the  other  extremity  C  ;  a  given  weight  AV  is  then  attached  to  P :  it  is 
required  to  determine  the  tension  of  the  string  when  in  equilibrium, 
as  also  how  much  of  the  cord  will  hang  below  the  ring,  (fig.  16.) 

The  tension  of  CP  is  measured  by  the  weight  W,  and  this  same 
tension  must  be  communicated  to  the  parts  CB,  BA,  AC,  since  the 
cord  passes  freely  through  the  ring  and  over  the  pullies ;  but  when 
three  equal  concurring  forces  are  in  equilibrium,  the  angles  formed 
by  their  directions  are  each  120°  ;  hence,  drawing  the  horizontal  line 
AD,  we  have  in  the  isosceles  triangle  ADC,  the  angles  A,  D,  each 
equal  to  30°,  and  the  angle  C  equal  to  120° ;  consequently,  as  the 
position  of  AB,  with  respect  to  the  horizon,  that  is,  the  angle  BAD, 
is  known,  we  know  in  the  triangle  BAC  the  side  AB  and  the  angles 
A  and  C  which  is  sufticient  for  the  determination  of  AC,  BC,  and, 
therefore,  of  the  place  of  C ;  also  the  perimeter  of  this  triangle  be- 
ing taken  from  the  whole  length  of  the  string,  leaves  CP  the  dis- 
tance of  the  weight  from  the  noose. 

If,  instead  of  the  loop  or  ring  at  C,  the  extremity  C  were  firmly 
fastened  by  a  knot  to  BCP  at  a  given  distance  from  the  other  ex- 
tremity P,  then  the  tension  of  CP  would  not  be  freely  communi- 
cated to  CA  or  CB,  but  whatever  tension  CA  had,  the  same  would 
be  communicated  to  AB  and  BC,  for  the  rope  being  freely  moveable 
over  A  and  B,  could  not  rest  so  long  as  either  of  these  tensions  pre- 
vailed. The  equal  tensions  of  CA,  CB,  as  also  the  position  of  the 
knot  C,  will  be  determined  in  this  case  as  in  problem  IV. 

Problem  VI. — The  extremities  of  a  given  cord  are  fastened  to  two 
hooks  given  in  position,  and  to  a  given  point  in  it  is  applied  a  power 
P  acting  in  a  given  direction :  to  determine  the  pressure  upon  the 
hooks  (fig.  17.) 

As  the  point  C  in  the  given  cord  ACB  is  given,  as  also  the  line 
AB,  therefore  the  three  sides  of  the  triangle  ABC  are  given,  and, 
consequently,  the  three  angles;  also,  as  the  direction  of  YCP  is 
given,  the  angles  at  Y  are  given;  hence  the  two  exterior  angles  ACP, 
BCP  are  given.  If,  therefore,  we  draw  CX  perpendicular  to  CY, 
the  angles  a,  a'  will  be  known,  so  that  calling  the  pressures  on  B 
and  k,  p  and/)',  we  shall  have,  by  the  conditions  of  equilibrium, 
p  cos.  a — jo'cos.  a'=0,p  sin.  a-fjo'sin.  a'=P, 

P  cos.  a'  Pcos.a' 

.'.  »= = 

COS.  a  sni.  a'-f-sin.  a  COS.  a'      sin.  ACB 

, P  COS.  a  P  COS  a 

COS.  a  sin.  a'  +  sin.  a  cos.  o'       sin.  ACB 
The  pressures/),/)',  therefore,  are  to  each  other  as  the  sines  of  the 


34  ELEMENTS  OF  STATICS. 

angles  .3,,  (3,  or  of  the  angles  ACP,  BCP;  but  this  much  we  might 
immediately  have  inferred  from  the  property  tliat  when  three  forces 
equiUbrate,  each  is  proportioned  to  the  sine  of  the  angle  between 
the  directions  of  the  other  two. 


CHAPTER  III. 

ON  THE  FUNICULAR  POLYGON  AND  CATENARY. 

(25.)  If  a  cord  be  kept  in  equilibrium  by  means  of  several  forces 
P,  Pj,  Pj,  P3,  &c.  acting  at  the  knots /j  j ,  p^,  p^,&,c.  the  figure  /),, 
p^,  Pj,  (fee  ,  which  it  forms  itself  into,  is  called  the  fiiniadar  po- 
lygon (fig.  18).  We  propose  here  to  investigate  the  conditions  of 
equilibrium  of  such  a  figure. 

And  first  it  is  obvious,  that  the  several  points  /J  j ,  7^2 '  ^^-  ^^^  ^^^^^ 
kept  in  equilibrium  by  the  three  forces  which  concur  there;  it  is 
equally  obvious  that  the  tension  of  the  string;;,,  yj^  being  the  same 
throughout,  there  is  the  same  pressure  upon  the  knot  /jj,  as  upon  the 
knot/)j,  but  exerted  in  the  opposite  direction,  and  the  same  of  any 
two  consecutive  knots /J^,  p^;  p^,  p^.  Sic.  Hence,  if  the  three 
forces  which  equilibrate  /),,  were  applied  io  p^,  the  equilibrium  of 
P2  would  remain  undisturbed;  but  of  the  forces  thus  acting  on  P2 
two  would  destroy  each  other,  since,  as  just  observed,  the  tension 
of /jj  P2  presses  the  points /),, /jj  ^^''th  equal  force  but  in  opposite 
directions ;  we  may,  therefore,  consider  but  four  forces  acting  on  p^ , 
viz.  the  forces  inp^  Pj?  and  in/jj  Ps  together  with  those  inpj  P, 
and  in /J,  P,.  Again,  if  the  four  forces  equilibrating /Jj'  ^^  trans- 
ferred to  /J3,  the  equilibrium  of  p^  will  remain  undisturbed,  and,  as 
before,  the  two  forces  due  to  the  tension  oi p^  p^  will  destroy  each 
other,  and  thus  the  point  pg  will  be  kept  in  equilibrium  by  five  forces 
of  which  only  one,  viz.  that  in  p^  p^  will  be  a  tension,  the  others 
being  the  forces  P,  PjjPj,  P3,  originally  applied  to  the  cord  at  the 
knots  p^,  P^^Pj.  Proceeding  in  this  way  from  knot  to  knot,  it  is 
plain  that  when  we  shall  have  arrived  at  the  last  knot  or  at  the  ex- 
tremity of  the  cord  that  the  point  will  be  kept  in  equilibrium  by  the 
concurrence  of  all  the  forces  originally  distributed  along  the  cord,  at 
the  points/jj,;j2'  «^^-  the  directions  of  these  forces  being  preserved. 

From  all  this  it  follows  then,  that  for  the  funicular  polygon  to 
exist,  the  intensity  and  directions  of  the  several  forces  acting  at  the 
knots,  must  be  such,  that  if  they  were  all  applied  to  one  point  they 
would  keep  it  in  equilibrium,  and  that  to  find  the  direction  and  ten- 
sion of  the  71  th  side  of  the  polygon  it  will  only  be  necessary  to  as- 


If  Off e  34  ■ 


-'^ 


FUNICULAR  POLYGON,  35 

certain  what  would  be  the  direction  and  intensity  of  the  resultant  of 
all  the  forces  acting  on  the  a — 1  preceding  knots,  if  they  were  all 
to  concur. 

It  thus  appears,  that  in  the  funicular  polygon  the  conditions  of 
equilibrium  are  the  very  same  as  if  the  forces  all  concurred,  or  were 
to  be  transferred  parallel  to  themselves  to  a  single  point,  so  that  if 
we  assume  three  rectangular  axes,  and  call,  as  before,  the  angles  at 
which  the  several  directions  of  the  forces  are  inclined  to  these,  a, 
ttj  ttj,  &c.  /3,  j3i,  ^21  ^^'  7'  72'  73'  <^^''  t^^^  conditions  necessary 
to  the  existence  of  the  funicular  polygon  will  be 

P  cos.  a  +  Pj  cos.    ttj-l-Pj  COS.  a2+&c.  =  01 
P  cos.  i3-i-Pi  COS.  /Si+Pg  COS.  132+ &c.  =  0  I    .   (1). 
P  cos.  y-j-Pi  COS.  Yi+Pa  cos.  yg-f  <^C-=0j 
When  the  forces  all  act  in  one  plane,  then  two  axes  taken  in  this 
plane  will  be  sufficient,  as  one  of  these  equations  then  becomes 
identically  0,  the  conditions  being 

P  COS.  a  +  Pi  cos.  ttj+Pa   COS.  a2+&C.=.0  )         ,^. 
P  COS.  /3  +  Pi  COS.  /3i  -fP2  COS.  13^  -i-&c.=0  5    '  ^'^■^' 

To  construct  the  polygon  in  any  particular  case,  we  must  know 
not  only  the  intensities  and  directions  of  the  several  forces,  but  also 
their  points  of  application  ;  when  these  are  known,  we  may  easily 
construct  the  successive  sides  of  the  polygon ;  thus  the  resultant  of 
P,  P^  being  determined,  we  shall  thence  have  the  direction  and  in- 
tensity of  the  force  in  p^  p^,  and  the  point /;2  being  known  we  thus 
have  the  side  p^  P2'i  i'^  ^^^^  manner  the  resultant  of  Pj  and  the  force 
in  /?2  Pi-,  just  determined,  will  make  known  the  intensity  and  direc- 
tion of  the  force  inp^,  p^,  so  that,  as  the  point  p^  is  given,  wemay 
construct  the  second  side  pg  Pa^  ^^^^  ^o  on  till  the  polygon  is  com- 
pleted. 

If  any  one  of  the  forces  were  attached  to  the  cord  not  by  a  fixed 
knot,  as  we  have  hitherto  supposed,  but  by  a  moveable  ring,  then  in 
the  equilibrated  state  of  the  system  the  tensions  on  each  side  of  the 
ring  would  be  equal,  and  would  therefore  form  equal  angles  with  the 
force  on  the  ring. 

In  this  way  may  the  absolute  tension  between  any  two  proposed 
knots  be  determined,  but  if  we  wish  merely  to  find  the  ratio  of  the 
tensions  of  any  two  sides  of  the  polygon,  then,  recollecting  that  each 
of  three  equilibrating  forces  acting  on  a  point  is  proportional  to,the 
sine  of  the  angle  between  the  other  two,  and  calling  the  several  ten- 
sions t,  t^,  t^,  &;c.  (see  fig.  18,)  we  have 

t sin.  Oj     t^     sin.  a^     t^     sin.  a^     . 

t^      sin.  a  '    ^2     sii^*  ^2^  ^3     sin.  04' 
Multiplying  these  equations  together,  and  omitting  the  factors  com- 
mon to  both  numerator  and  denominator,  we  have  generally 


39  ELEMENTS    OF    STATICS. 

t      sin.  a.  sin.  a,  sin.  a,  .  .  .  .  sin.  ff,„_i 

—  =  -. —. ^—. : ....   (3). 

/"  sm.  a  sin.  Oj  sm.  a^  .  .  .  .  sin.  021,-2 
Should,  therefore,  the  zngXes  a,  a^,a^.,  &Lc.he  equal  throughout, 
the  tension  will  be  uniform  throughout,  and,  conversely,  if  the  ten- 
sion be  uniform  throughout,  the  angles  must  be  all  equal ;  when, 
therefore,  the  angles  are  equal,  tlic  uniform  tension  of  the  cord  is 
measured  by  either  of  the  extreme  forces  P,P„  ,  wliich  are  necessarily 
equal  in  intensity,  as  they  measure  the  equal  tensions  of  the  extreme 
sides  of  the  polygon. 

(26.)  Tlie  most  important  case  of  the  funicular  polygon  is  that  in 
which  the  several  forces  P,,  Pj,  &lc.  (fig.  19,)  are  weights,  acting  in 
the  same  vertical  plane  upon  fixed  points  of  the  cord  when  suspend- 
ed at  its  two  extremities  P,  P„  ,  (fig.  19),  we  shall  therefore  consider 
this  case  in  particular,  and  first  we  may  remark,  that  the  polygon  so 
formed  will  lie  wholly  in  the  vertical  plane  of  the  forces,  for  of  the 
three  equilibrating  forces  concurring  in  /),,  two,  viz.  those  in;>,, 
P,  and  in  p.^  Pj,  are  in  the  vertical  plane;  therefore  the  third,  or 
that  in 7) J, 7^2'  n^ust  be  in  the  same  plane  ;  also,  this  last  and  that  in 
p^  P2  being  in  the  vertical  ])lane,  the  force  in/^j  P^  must  be  in  that 
plane,  and  so  on.  Let  us  then  draw  in  this  plane  the  horizontal  and 
vertical  axes  PX,  PY  ;  the  angles  aj,  Oj,  &c.  which  the  directions 
of  the  forces  make  with  the  first  of  these  axes,  are  each  equal  to  90°, 
and  the  angles  /i,,  iSj,  &;c.  made  with  the  other  axis,  are  each  0; 
hence,  denoting  the  sum  of  all  the  weights  Pj,  P,,  &c.  by  AV,  the 
equations  of  equilibrium  (2)  at  page  35  become,  in  this  case, 

PCOS.  a+PnCOS.  a„=0  ~>  ,.. 

Pcos./3  +  P„cos.  ^„4-W=0  5  •••'K^h 
where  P  and  P„  are  the  pressures  on  the  two  points  of  suspension  ; 
these  pressures  are  therefore  readily  determinable  if  the  angles  a. 
On ,  that  is  the  directions  of  the  extreme  cords  are  given,  there  beinw 
no  necessity  to  know  the  situation  of  the  knots,  nor  yet  the  separate 
forces  P,,  P,,  &c.,  but  only  their  sum  W.  All  this,  indeed,  may 
be  determined  from  the  equations  themselves  ;  thus,  let  t  represent 
the  tension  of  any  side  of  the  polygon,  and  let  us  put  a  for  the  angle 
it  makes  with  the  horizontal  axes,  and  b  the  angle  it  makes  with  the 
vertical  axis. 

The  tension  t  may  be  considered  as  exerting  a  pressure  upon  the 
knot  at  that  extremity  of  the  proposed  side  which  is  farthest 
from  the  point  P,  so  that  substituting  this  pressure  for  P„  in  the 
equations  (1)  and  calling  the  sum  of  the  weights  between  P  and  this 
knot  w,  we  have 

P  cos.  a  +  ^  COS.«  =  0  )  ,    , 

P  COS.  fi-\-t  COS.  b-\-w=Q  ^    .  .  .  .  (.-ij, 
two  equations  from  which  the  two  unknowns  t  and  a  may  be  deter- 


THE    CATENARY.  37 

minea,  and  thus  the  intensity  and  direction  of  the  force  in  any  side 
of  the  polygon  ascertained ;  and,  from  knowing  the  intensities  and 
directions  of  the  forces  in  two  adjacent  sides,  Ave  find  the  intensity 
of  the  vertical  force  at  the  angle  by  taking  the  resultant. 

As  to  the  ratio  of  any  two  tensions,  it  is  involved  in  the  general 
expression  (3)  of  last  article,  Avhich,  because  in  the  case  under  con- 
sideration a^  is  the  supplement  of  a2,  ci^  the  supplement  of  a^,  and 

,  ^     t      sin.  02  71-1  •     1-1  ^      sin.  flfon'-i 

so  on,  reduces  to  — = —- in  like  manner  —  = r^ , 

tn         sin.  a  In'  sin.  a 

.•.  —  =^-^ — ^nL  •  so  that  the  tensions  of  any  two  sides  of  the  poly- 

tji       sin.  (l2n' — 1 

gon  are  reciprocally  as  the  sines  of  the  angles  loliich  they  forvn, 
ivith  the  vertical  axis. 

Since  these  angles  are  the  complements  of  those  which  the  same 
sides  form  with  the  horizontal  axis,  we  may  substitute  the  cosines 

of  these  latter  for  the  sines  of  the  former,  or  because  cos.= we 

sec. 

may  say  that  the  tensions  are  directly  as  the  secants  of  their  incli- 
nation to  the  horizon. 

(27.)  TTie  Catenary.  By  referring  to  equations  (1),  last  article, 
we  see  that  they  express  the  conditions  of  equilibrium  of  these  three 
forces  acting  at  their  point  of  concurrence,  viz.  the  force  P  inclined 
at  an  angle  a  to  the  horizon,  the  force  P^  inclined  at  an  angle  an ,  and 
the  vertical  force  W.  Hence,  in  our  polygon  (fig.  19),  if  we  pro- 
duce the  directions  of  the  pressures  P,  P„,  that  is  the  extreme  sides 
of  the  polygon,  the  point  O  in  which  they  meet  must  be  the  point 
of  concourse  of  which  we  speak,  at  which  the  vertical  weight  W 
and  the  pressures  P,  P„  acting  maintains  the  equilibrium  of  O  ;  these 
pressures  are  therefore  the  same  as  if  all  the  weights  acting  at  the 
angles  of  the  polygon  were  collected  and  applied  at  O,  it  would 
therefore  not  be  improper  to  consider  W  so  applied  as  the  resultant 
of  all  the  original  weights. 

If  we  suppose  in  our  polygon  the  weights  to  be  attached  at  equal 
distances,  the  less  these  distances  are  taken  the  greater  will  be  the 
number  of  sides  of  the  polygon,  and,  consequently,  the  figure  Mali 
approach  the  more  nearly  to  a  curvilinear  form,  which  form  it  must 
actually  assume  when  the  distances  between  two  consecutive  weights 
become  0,  that  is,  when  the  weights  act  upon  every  point  of  the  cord  ; 
now  this  is  the  same  as  considering  every  point  itself  to  be  weighty, 
so  that  the  curve,  of  which  we  speak,  will  be  that  which  a  perfectly 
flexible  physical  line  or  chain  actually  assumes,  when  suspended 
at  its  extremities.     It  is  called  the  catenary  curve,  (fig.  20.) 

The  direction  of  the  pressures  on  the  points  of  suspension,  will, 
obviously,  be  tangents  to  the  catenary  at  those  points,  and,  from 
D 


38  ELEMENTS    OF    STATICS. 

what  has  been  said  above,  it  appears  that  the  point  O,  in  which 
these  directions  meet,  would  be  kept  in  equilibrium  by  the  pressures 
or  the  tensions  of  the  lines  OP,  0P„ ,  and  by  the  whole  weight  W 
of  the  chain  suspended  at  O. 

(28.)  Let  us  seek  the  equation  of  the  catenary,  supposing  that 
the  cord  or  chain  is  uniformly  heavy  throughout,  that  is  that  the 
lengths  of  any  two  portions  are  to  each  other  as  their  weights.  Ta- 
king the  horizontal  and  vertical  lines  PX,  PY,  for  axes  of  co-ordi- 
nates, we  shall  have  for  any  point  M,  Vm=x,  mM=^y  and  PM=5  ; 
and  the  portion  s  of  the  cord  is  held  in  equilibrium  by  the  tensions 
at  P  and  M,  acting  in  the  directions  OP,  OM  of  the  tangents  at  those 
points  and  also  by  the  weight  of  s;  or  the  point  O  is  held  in  equi- 
librium by  the  same  tensions  and  the  vertical  weight  s,  the  relation 
therefore  between  s  and  the  tensions  is  determined  by  the  relation 

sin.  POM      s 
between  the  sines  of  the  angles  about  0,  that  is  (14)-.  '.\  =—, 

where  p  represents  the  pressure  on  P.     Now 
sin.  POM=sin.  (lOM-j-POI) 

=sin.  lOMcos.  POI-1-cos.IOMsin.  POl 
=sin.  7nM0  sin.  a — cos.  mMO  cos.  a. 

consequently,  since  mM0=M05 

sin.  POM. 

—■ — ,,,^    =sm. a»— cot. mMO  cos.  a ; 

sin.  MOs 

but  (Bi^.  Calc.  p.  113,)  cot.  mMO=-^  hence 

s       .  dy  dy  ... 

— =sin.a -r-  cos.  a  .•.  S=^p  sin.  a — p  -^  cos.  a  . . .  ( 1  J, 

p  dx  ^  ^  dx  ^  ^ 

which  is  the  differential  equation  of  the  catenary. 

We  may  obtain  a  differential  equation,  involving  only  x  and  y, 

provided  we  differentiate  this  with  respect  to  x  and  substitute  for 

IL  its  equal  -^  =1  1  _,.  ^^  for  ^^e  thus  have 

d^y 
dx^ 


■^d^=-p'''-''d^''-'==-p'''-\\'';^ 


4 


Jdij<'  dHj 

1  -f  -r^  =  —  pC0S.o-r4 

dx^ 

The  numerator  of  the  fraction  in  the  second  member  of  this  equa- 
tion will  obviously  be  the  differential  of  the  denominator  if  we  mul- 
tiply it  by  dy  ;*  by  doing  this,  therefore,  and  then  integrating  the 

*  This  is  the  same  as  multiplying  both  sides  by  ~  and  then  multiplying  by  dx, 

dx 

to  prepare  each  side  for  integration. 


THE  CATENARV  39 


equation,  we  have  y=. — p  cos.  a  ^  1  +  -~  +  c,  from  which  we 

dy       \/  (c — y)^ — o'^cos.^a 
get  -^  = — ! ^ i- ....  (2). 

°        dx  pCOS.  a  ^    ■" 

It  remains  to  determine  the  constant  c,  for  which  we  have  this  con- 
dition, viz.  that  when  x=0  and2/=0,  that  is  at  the  point  P,—r-  = 
tan.  a,  so  that  for  this  point  the  equation  (2),  just  deduced,  is 


sin.  o     \/(c^  —  w^  cos.*^  a) 

tan.  a= =^-^ i-, 

cos.  a  p  cos.  a 


orjjsm.  a=v'c3 — p^cos.^'a 

.'.  c^=p'^  (a'm.^  a  +  cos. 2  a)  =p^  .'.  c=p  : 
hence  the  differential  equation  of  the  catenary  (2)  is 

dx  pcos.a  ^ 

dym 
If  we  substitute  this  value  of-;—  in  the  equation  (1),  there  results 


s=p  sin.  a— '^(  p — y)^ — p^cos.^a.  (4),  which  expression,  for  the 
length  of  the  arc,  proves  that  it  is  rectifiable. 

In  order  to  obtain  an  equation  between  x  and  y,  independently  of 
differentials,  let  us  put  in  (B)  p — 2/=-,  p  cos.  a=a,  then  dy= — 
dz  and  the  equation  reduces  to 

dz 
dx= —  a  — —  .  .  .  (5) ;  to  render  this  rational  we  must  as- 

k/  z^  —  a^ 

sume  x/^TZr^  =  z — z',  from  which  we  get  the  equation  2zz'= 

a^-\-z'^,  which  differentiated  gives 

zdz'4-z'dz:=  z'dz'r. = :-=  —  d  log.  z'; 

_  z—z'  z'  ° 

that  is,  from  (5)  dx=ad  log.  z'  .••  x=a  log.  z'-\-c  that  is  restoring 
the  value  of  z',a,'=a  log.  f  z  —  .y/z^  —  a* |  -fc ;  or,  restoring  the  values 
of  z  and  a 


a7=  ;j  COS.  a  log.  [(p  —  y)  —  \/{p—yY — jo'^cos.^al+c  (6). 
The  constant  c  may  be  determined  from  the  condition  that  a;=0 
when  .V=0,  the  origin  being  at  P,  this  condition  gives 
c= — p  cos.  a  log.  {p{\  — sin.  a)  I ,  and  thus  the  equation  (6)  is 

1        <(P—y) — v^V^  — 2nv-fP^sin.2a,  ,  , 

2;=  »  cos.  a  log.  \— — ^ ^ ^-^  '  ^ \  .  .     Cj^ 

^  ^   ^  p(l— sin.  a)  ^        ^  ^ 

In  order  to  determine  the  lowest  point  in  the  catenary,  or  that  point 
at  which  the  tangent  of  the  inclination  to  the  horizon  is  0,  we  must  put 


40  ELEMENTS  OF  STATICS. 

fly 

-p=0  in  (3),  which  will  give  for  y  the  value  3/=/)  (1  —  cos.  a)...  (8), 

und  this  put  for  y  in  the  equation  (7)  gives 

x=p  cos.  a  log. '-. .  .  .  (9) ;  also  the  length  s  of  the  cord 

1  —  sin.  a  ^ 

hanging  between  the  point  of  suspension  P  and  the  lowest  point  is, 
by  equation,  (4)  s=p  sin.  a  .  .  .  (10). 

(29.)  If  botli  points  of  suspension  P,  V  „,  are  in  the  same  hori- 
zontal line,  the  portion  of  the  cord  between  P  and  the  lowest  point 
will,  obviously,  be  equal  in  length,  and  symmetrical  in  figure  to  that 
between  P„  and  the  lowest  point:  hence  the  value  of  s  in  (10)  will 
be  half  the  length  of  the  cord.  When,  therefore,  we  know  the 
horizontal  distance  D  of  the  points  P,  P„  and  the  length  L  of  the 
cord,  we  may  by  help  of  the  last  three  equations  determine  the  angle 
a  and  the  tension  p  :  thus  dividing  (9)  by  (10)  we  have 
D     COS.  o ,  COS.  o  ,,,^ 

T=— loff-  -; ■■ ; ....  (11) 

L      sm.  a     °     1 — sm.  a  ^     ^ 

an  equation  from  which,  the  unknown  quantity  a,  may  be  determined 

by  approximation,  after  which/)  will  be  given  by  (10),  viz. 

knowing,  therefore,  o  and  p,  we  may  readily  find  the  tension  t  at 

any  point  of  the  cord;  for  if  s  be  the  length,  hanging  between  P 

and  this  point,  then  from  equations  (2)  art.  26 
p  COS.  a-\-t  COS.  a=0        1 
p  COS.  ji-i-t  COS.  6  +  s=0  l.  .  .  (13) 

also  cos.^  a-\-cos.^  b^l        j 

from  which,  cos.  a  and  cos.  b  being  eliminated,  there  results  for  t 

the  value  t=^\/p'^  —  "Ipn  sin.  o  -f-*^ 


=  v7j^cos.'^a+(/>  sin.a  —  sy  ....  (14). 
This  expression  we  may  simplify  and  render  independent  of  p  ; 
thus,  substitute  for  s  the  value  in  equation'(lO),and  we  shall  thus 
have  for  the  tension  a  at  the  lowest  point  A, 

a=p  COS.  o=,  (equa.  12),  5  L  cot.  a  .  .  .  (15), 
consequently,  by  substitution,  the  general  expression  for  ^  is 

f=^|L^cot.''a+(iL  — s)»  ....  (16) 
and  from  this  we  may  get  the  value  of  cos.  o,  by  means  of  the  first 
of  (13).  It  thus  appears  that  when  a  flexible  cord,  or  chain  of 
given  length,  is  suspended  from  two  points,  at  a  given  distance  from 
each  other,  in  the  same  horizontal  line,  we  may  always  determine 
its  tension  and  direction  at  any  point. 

(30.)  The  general  equations  of  the  curve  (4)  and  (7),  as  also  the 
expression  (14)  for  the  tension  at  any  point,  will  become  simpler  in 


THE    CATENARY.  41 

form,  if  the  origin  of  the  axes  be  taken  at  tne  lowest  point  A  of 
of  the  curve ;  for  confining  ourselves  to  the  consideration  of  the 
branch  A  P„,  we  may  view  A  and  P„  as  the  two  points  of  suspen- 
sion of  the  cord  A  P„,  in  Avliich  case  a=0,  and  as  y,  Avhich  has 
heretofore  been  measured  downwards,  will  now  be  measured  up- 
■wards,  we  must  change  the  sign  which  it  carries  in  the  preceding 
formulas  when  we  wish  to  adapt  them  to  this  arrangement  of  the 
axis.  Calling  the  tension  at  A,  a,  vie  thus  have,  by  equation  (4), 
s=,y2ay=y^  ....  (1),  also  from  equation  (7) 

or  since  by  the  equation  just  deduced 

^  a^-\-s'=^  a"-+2ay+y"'=  a+y, 
we  may  put  the  last  equation  under  the  form 

x=a\og.\ \.  .  .  .  (3). 

The  expression  (14)  for  the  tension  t  at  any  point  will  be 
t=^  d'+s''  ....  (4). 

All  these  equations  involve  the  unknown  tension  a,  but  this  may 
be  determined  by  trial  from  (3),  since  we  know  the  values  of  .r  and 
s  in  one  case,  viz.  .r=iD  and  S:=5L.  By  means  of  this  value 
of  s  and  a,  thus  determined,  we  may  obtain  y,  that  is  the  length  of 
A  A',  or  the  distance  of  the  origin  A  from  the  middle  of  P  P„,  and 
knowing  thus  the  position  of  the  axes,  the  curve  may  be  con- 
structed from  its  equation  (2). 

We  shall  now  give  an  example  of  the  preceding  formulas. 

Problem  I. — (31.)  The  length  of  a  heavy  flexible  chain  is  just 
double  the  horizontal  line,  joining  the  points  of  suspension  to  de- 
termine the  pressure  on  these  points,  the  inclination  a  to  the  hori- 
zon, &c. 

It  is  obvious,  from  equation  (11,  p.  40),  that  it  is  suflicient  to 
know  the  ratio  of  the  length  L  to  the  distance  D,  in  order  to  de- 
termine the  angle  a.  We  shall  not,  however,  employ  this  formula, 
but  that  above  marked  (3),  as  this  is  of  more  easy  application.  If 
then,  in  this  formula,  we  suppose  s  equal  to  half  the  length  of  the 
chain  equal  to  1,  then  x,  which  is  half  the  horizontal  distance  be- 
tween the  points  of  suspension,  must  be  i ;  moreover  a  will  then 
be  cot.  a  (equa.  15,  p.  40) :  hence  we  shall  have 

a  =  «  log.  { ^1 ;  the  logarithm  here  indicated  being  hy 

perbolic,  it  will  be  convenient  to  convert  it  into  a  common  loga- 
D  2  6 


42  ELEMENTS    OF    STATICS. 

rithm  which  requires  that  we  multiply  its  value  by  '43429,  so  that 
•21715==alog.  f^tl_ifLl_i|.     Now   the  first   side  being  little 

more  than  |,  a  near  value  of  a  at  once  presents  itself,  viz  a=:^ 
^•2,  which  substituted  in  the  second  member  gives  '2  log.  10-099 
=•20086;  a  result  which  is  rather  too  small;  let  us,  therefore, 
take  a  a  little  larger,  making  it  a=i="25;  the  second  member  will 
then  be  k  log.  8- 1231 =-22743,  which  is  a  little  greater  than  the 
true  result.  Hence  by  the  known  rule  of  trial  and  error,  since  the 
differences  of  the  results  are  nearly  as  the  differences  of  the  sup- 
positions which  have  led  to  them,  we  have 
•22743=-20086=-02657  :  -22743— •21715=-01028  :  :  -05  :  -0194 

.•.a=-25— •0194  =  -2306  nearly. 
To  obtain  a  still  nearer  approximation  to  the  truth,  let  us  now  as- 
sume a=-23,  then  the  second  member  of  the  equation  in  a  is 
•23,  log.  8-8092=-21735,  and  as  this  is  a  little  too  great,  let  us, 
finally,  put  a=-22  and  we  have  -22  log.  9-1996=-21203 ;  conse- 
quenUy,         -21735— •21203=-00532  :  21735— -21715 

=  -0002  :  :  -01  :  -00037,  .-.  a=-23— -00037=-22963 
Seeking  now  in  the  tables  for  the  natural  cotangent  correspond- 
ing to  this  number,  Ave  find  for  the  angle  a,  or  the  inclination  of 
the  chain  to  the  horizon  at  cither  point  of  suspension,  the  value 
o=77°,  4',  therefore  putting  1=^  L,  the  pressure  on  these  points  is 

(equation  12),  »=-: = --;   and   the  tension  at  the  lowest 

^  '  "^     sm.  tt     -97463 

point  A  is  (equation  15)  a=l  cot.  o= -22963/ 

Also  for  the  distance  of  A,  below  the  horizontal  line  P  P„,  we 

have  (equation  8),  2/=p  (1  —  cos.  a)  =  -79638/.  the  depth  of  th$ 

lowest  or  middle  point. 

Problem  II. — A  heavy  flexible  chaui,  100  feet  in  length  and 
weighing  1000  lbs.  is  suspended  at  its  extremities  to  two  fixed 
points,  in  the  same  horizontal  line,  95  feet  H  in.  asunder.  It  is  re- 
quired to  determine  the  greatest  depth  of  the  curve,  the  tension  at 
the  lowest  part  and  the  tensions  at  the  points  of  suspension. 

In  this  example  the  ratio  ^  is  -95125;  hence,  as  in  the  former 

problem,  we  shall  have  to  determine,  a,  or  rather  cot.  a,  from  the 
equation 

•43429x-95125=-41312=alog.  fii^^'-tii. 

a 

After  a  little  consideration  we  find  that  c:=|=l-75  is  a  near 

value,  substituting  this,  therefore,  in  the  last  member,  we  have 

|log.  1*7232 =-4 1359;  this  being  a  little  greater  than  the  true  re- 


THE    CATENARY.  43 

suit,  let  US  take  a?= 1-7,  and  we  then  have  1-7  log.  l-7484=-41249  ; 

consequently, 

•41359— •41249=-0011  :  41359— 41312  =  -00047  :  :  -05  :  -02136 

.-.  a=:l-75— •02136=1-72864  nearly. 
Again,  let  a=l-73,  then  1-73  log.  1-7331  =-41316. 
Comparing  this  result  with  that  obtained  by  the  first  supposition, 
we  have  the  proportion 
•41359  — •41316='00043   :  -41359- •41312=-00047.-:-02:-02219 

.-.  a=l-75— -02219  =  1-72781, 
this  number  corresponds  to  the  natural  cotangent  of  30°,  4',  there- 
fore, for  the  inclination  a  we  have  a=30°,  4' 

/  50 

.••  pressure,  a= = =99-8  ft.=  998  lbs.* 

^  -^      sm.a     -50101 

also  tension  at  A,  a=/  cot.  a=50x  1*72781  =86-4  ft.=8641bs.  and 

distance  of  A  from  PP„,  y=p  (1 — cos.  tt)  =  13-4  ft. 

Problem  III. — A  chain  of  given  length,  2  /  hangs  freely  over  two 
given  points,  in  ahorizontal  line,  in  what  position  will  itrest?(fig.  21). 

When  the  chain  is  at  rest  it  is  plain  that  what  in  the  former  pro- 
blems was  the  pressure  upon  the  points  of  suspension  P,  P„ ,  will 
here  be  equivalent  to  the  weight  of  either  PP' or  of  PkP„'  the 
parts  hanging  vertically,  these  parts  are,  therefore,  equal  to  each 
other,  and  to  what  we  have  hitherto  called  p ;  hence,  calling  half 
the  length  of  the  catenary  s,  the  expression  for  I  will  be  (equa.  10, 
p.  40,)  l=:s-^p=p  (sin.  a-fl)  ...  (1)  ;'  also  the  expression  for  half 

COS    (X 

PPn  or  /'  is  (equa.  9),  /'=pcos.  alog. ': ....  (2)  ;  divi- 

l"^— sin.  a 

ding  this  by  the  last  equation  we  have 

I' cos.  o       .  cos.  o  sin.  j3  sin.  |3 

/        1-f-sin.  a       °"  1 — sin.  tt  ~  1 -f  cos.  i3      °'l  •—cos.  /3 

=tan.  1  i3  log.^^j^^-^t  =  —  tan.  i  /3  log.  tan.  1  /3 ; 

the  first  member  of  this  equation  being  given,  it  follows  that  to 
determine  /3,  the  angle  at  which  the  chain  is  inclined  to  the  vertical, 
we  have  only  to  find  by  trial  a  number  such  that  when  multiplied 
by  its  logarithm,  the  product  shall  be  equal  to  a  given  number. 


*  Because  the  weight  of  a  foot  of  the  chain  is  10  lbs. 

xppj  sin,  g  \/l  —  COS.23    I]  — cos.  0 

1  +  cos.  ^  1  +  cos.  o  \  1  +  cos.  /S  —^^'  i  ^ 

Cyoutig^s  Trigonometry,  p.  37.)     In  like  manner, 

sin,  a  _     II  +  cos.  s  _  1 

1  —  cos.  g,      ~\\  -fToT^        tan.  ;^  '^ 


44  ELEMENTS    OF    STATICS. 

When  this  is  found,  a  becomes  known,  and  from  the  above  equa- 
tion (1),  we  getp=  — — -,  which  gives  the  length  of  that  part 

sm.  a  ^  1 

of  the  chain  which  hangs  vertically  on  each  side  of  the  curve. 

Problem  IV. — Given  the  distance  2/'  between  two  fixed  points  in 
the  same  horizontal  line,  to  determine  the  length  of  the  shortest 
chain  that  can  remain  suspended,  as  in  the  preceding  problem. 

We  have  just  seen  that  -j= — tan.  \  ,3  log.  tan.  A  /3 ;  and  as  /  is  to 

be  a  minimum,  -j  must  be  a  maximum,  /'  being  constant,  that  is  to 

say,  calling  tan.  i  ^,  x,  —  x  log.  x= — log.  x^  =log.  —  =  max.  .'.— 

x^  X 

max,  or  x^  =min. 

This  equation  is  solved  at  p.  74  of  the  Differential  Calculus,  where 

the  value  of  x  is  found  to  be  a;= — = =tan.  i  3 

e      2-7182818...  -• 

.-.  cotan.  i  ,3=2-7182818...,  .-.  |  ,3=20°,  12'  .-.  /3=40°,  24'. 

It  appears,  therefore,  that  in  this  case  we  must  have 

-r= log.  -i=-  .-.  /=cZ'=2-7182818/', 

/  e      °    e       e 

so  that  if  the  distance  21'  between  the  fixed  points  be  10  feet,  then 

the  length  2l  of  the  shortest  chain,  which  will  suspend  itself  by 

hanging  over  them,  will  be  27'182818  feet. 

For  the  length  of  each  part  of  the  chain  hanging  vertically  we  have 

p=-^ = = =1  I  (1  +tan.''  i  ^) 

'       sin.  tt+l       cjs.,3-(-i       2cos.aA/3      ^     ^    "r  2  f-y 

=}  /  (IH — -),  and  for  the  distance  of  the  lowest  point  in  the  cate- 
nary from  the  horizontal  line 

y=p  (I  —  COS.  a)=p  —  I r^ r-.p  —  /  tan.  i/3 

^     ^  ^  ^    ^  l+sm.  a      ^  ^ 

For  further  particulars  respecting  the  curves  formed  by  flexible  lines, 
acted  on  by  different  forces,  as  also  respecting  those  which  elastic 
lamina;  assume  under  like  influences,  we  must  refer  the  student  to 
Professor  JVhewelVs  Mechanics,  chap.  x.  and  xi.,  (the  first  edition 
of  this  work  is  here  referred  to,)  where  these  matters  are  very  elabo- 
rately treated,  and  at  great  lenerth. 


ECIUILIBRIUM    ON    A   SURFACE.  45 

CHAPTER  IV. 

ON   THE    EQUILIBRIUM   OF    A    POINT    ON    A   CURVE    OH   SURFACE. 

(32.)  If  a  material  point  be  placed  upon  a  curve  surface,  and  be 
kept  in  that  place  by  the  mutual  action  of  any  number  of  forces  ap- 
plied to  it,  the  resultant  of  these  forces  must  be  in  the  direction  of  the 
normal  to  the  surface,  and  must  be  equivalent  to  the  pressure  whicli 
the  surface  sustains.  For,  if  the  resultant  had  any  other  direction, 
we  might  decompose  it  into  two,  one  in  the  direction  of  the  normal, 
and  the  other  in  the  direction  of  a  tangent  to  the  surface ;  the  first 
of  these  would  be  opposed  by  the  resistance  of  the  surface,  but  the 
second,  being  unopposed,  would  cause  the  point  to  move.  Consider- 
ing, therefore,  the  resistance  which  the  surface  opposes  to  the  nor- 
mal force,  as  one  of  the  system  of  forces  acting  upon  the  proposed 
point,  we  may  altogether  disregard  the  surface  and  view  the  point 
as  a  free  point  kept  in  equilibrium  by  the  system  of  forces  P,  Pj, 
Pj,  &c.  and  N  ;  and,  hence,  we  have  the  same  equation  of  condition 
as  in  (22),  that  is  putting  6,  0',  9",  for  the  angles  which  N  forms 
Avith  three  rectangular  axes,  we  have 

N  cos.  9   -f  P  cos.  a  -f-  Pi  COS.  ai  +  Pg  cos.  Og  -f  &c.=0 
N  COS.  9'  -j-  F  cos.  i3  -f  Pj  cos.  ^^  -j-  P^  ^^^-  ^'2  +  &c,  =  0 

N  COS.  (9"-f-  P  COS.  y  +  Pi   cos.  yi    +  P2   ^^S.  y^    -f   &C.=0  ; 

or,  putting  as  at  (22)  X,  Y,  and  Z,  for  the  sums  of  the  components 
along  the  respective  axes,  the  three  equations  may  be  written 
N  cos.  Q  -f  X=0,  N  cos.  9'  +  Y=0,  N  cos.  9"+Z=0  .  (1) 
Now  we  must  here  remark  that  the  angles  6,  9',  9",  which  de- 
termine the  direction  of  the  force  N,  are  entirely  dependent  on  the 
equation  of  the  surface,  and  on  the  co-ordinates  of  the  point  to  which 
the  forces  P,  P,,  &c.  are  applied.  Knowing,  therefore,  the  equa- 
tion of  the  surface,  and  the  position  of  the  point,  we  may  always 
determine  the  direction  in  which  the  resultant  of  the  applied  forces 
P,  Pi,  &c.  must  necessarily  act  to  ensure  the  equilibrium.  Thus, 
the  equation  of  the  surface  being  m=F  (x,  y,  z,)=0,  we  have  {Bif. 

Laic.  p.  166,)  COS.  9=u  ^-,  COS.  9  =v-7-»cos.  9   =v-7-, 
ax  ay  ciz 

where  v=- 


dx^'^dy'^'^dz^' 


The  values  of  cos.  9,  cos.  6',  cos.  9",  being  thus  found,  if  we  sub- 
stitute them  in  the  equations  (1)  and  then  eliminate  N  between  each 
two,  we  shall  obtain  two  equations  expressing  the  conditions  which 
must  exist  among  the  applied  forces  and  their  inclinations  to  the 


46  ELEMENTS  OF  STATICS. 

axes,  in  order  that  their  resultant  may  be  a  normal  force.     Indeed, 
to  obtain  these  equations,  we  need  not  take  the  trouble  to  first  cal- 

1  ,  1        .     •        du  du        ,  dii   .  ,     ,, 

culate  V,  but  may  simply  substitute  -:-,  -;-,  and  -7-»  for  6,6  ,6  ,rc- 

dx  dy  dz 

spectively,  because  v  disappears  with  N.  It  hence  appears,  that 
by  means  of  the  equation  of  the  surface,  and  the  position  of  the 
point,  the  equations  of  equilibrium,  originally  three,  become  re- 
duced to  two.  If,  indeed,  one  of  the  axes  of  reference,  as  the  axis 
of  z,  coincide  with  the  normal,  and,  consequently,  originate  at  the 
point,  then  to  find  these  two  equations  it  will  not  be  necessary  to 
know  the  equation  of  the  surface;  for  then  the  equations  (1)  will  be 
P  cos.  a-j-Pj  cos.  aj-f  P2  cos.  02-f<S:c.=0 
Pcos.  /3  +  Pi  cos.  /3i-f-P2  COS.  /32  +  &c.=0 
N  +  P  COS.  v+Pi  COS.  71  +P2  COS.  72  H-&c.=0  : 
the  two  first  of  which  are  all  that  are  requisite  to  establish  the  equi- 
librium, for  if  these  hold,  the  third  must  necessarily  hold,  since  the 
intensity  of  the  normal  forces  or  pressure  may  be  any  whatever, 
without  disturbing  the  equilibrium.  This  third  equation  is  neces- 
sary, however,  to  determine  the  resultant  of  the  applied  forces,  or 
the  intensity  N  of  the  whole  normal  pressure. 

(33.)  Having  thus  briefly  noticed  the  conditions  necessary  for 
the  equilibrium  of  a  point  on  a  surface,  viewing  the  matter  in  the 
utmost  generality,  we  shall  now  consider  the  equilibrium  under 
more  particular  circumstances,  taking  those  cases  only  which  are 
likely  to  present  themselves  in  nature,  w'here  the  acting  forces  are 
gravity  or  weight. 

In  this  point  of  view,  where  we  are  to  consider  the  means  of  re- 
taining a  heavy  point  on  a  given  surface,  the  problem  becomes  very 
much  simplified  from  the  following  considerations.  A  heavy  body 
(considered  as  a  point)  placed  upon  a  surface,  will,  unless  it  presses 
entirely  in  the  direction  of  the  normal,  tend  to  move  on  that  surface 
towards  the  horizon,  and  this  tendency  will  be  in  a  certain  determi- 
nate direction,  viz.  in  that  direction  which  is  nearest  the  perpendicu- 
lar to  the  horizon. 

Now  if  through  the  point  at  which  the  body  is  placed  a  tangent 
plane  to  the  surface  be  drawn,  and  from  the  same  point  a  tangent 
line,  perpendicular  to  the  horizontal  trace  of  the  tangent  plane, 
this  tangent  line  will,  obviously,  be  shorter  than  any  other  drawn 
from  the  same  point  to  the  same  trace ;  hence  this  tangent  line  must 
be  more  nearly  perpendicular  to  the  horizon  than  any  other  through 
the  body's  plane,  and,  consequently,  in  the  direction  of  this  line 
the  body  will  tend  to  move ;  the  very  same  conditions,  therefore, 
which  would  be  necessary  to  counteract  the  tendency  to  move  on 
the  surface,  would  be  necessary  to  counteract  the  tendency  to  move 


EQCILIBRIUM  ON  A  SURFACE.  47 

if  the  point  were  placed  on  this  line,  instead  of  on  the  surface ;  we 
may,  therefore,  in  seeking  the  conditions  of  equilibrium,  substitute 
the  straight  line  of  which  we  are  speaking,  for  the  curve  surface. 
The  position  of  this  line  is  always  determinable  from  the  equation 
of  the  surface,  for  the  trace  is  found  by  putting  in  the  equation  of 
the  tangent  plane,  2=0,  supposing  the  plane  oi  xy  to  coincide  with 
the  horizon,  and  the  line  sought  will  be  represented  by  the  equations 
which  characterize  the  perpendicular  to  this  trace  through  the 
given  point.  In  the  vertical  plane  through  this  line  must  the  forces 
act,  so  that  we  may  resolve  each  into  two,  one  acting  in  this  line, 
and  the  other  in  a  line  perpendicular  to  it;  the  sum  of  the  forces 
acting  in  this  latter  line,  be  it  what  it  may,  will  be  counteracted  by 
the  resistance  of  the  line,  so  that  to  establish  the  equilibrium  it  will 
merely  be  necessary  that  the  sum  of  the  forces,  acting  in  the  in- 
clined line,  be  0. 

Hence  we  shall  need  but  one  equation  of  condition ;  and,  indeed, 
in  whatever  two  rectangular  directions  the  forces  be  resolved,  the 
conditions  of  equilibrium  will  always  be  expressed  in  a  single  equa- 
tion. For  calling  the  resistance  of  the  line  R,  and  the  sum  of  the 
components  of  the  other  forces  X  and  Y,  we  know  that  the  condi- 
tions of  equilibrium  are 

P     X     _     p 
CX+Rcos.  a=0)        J    cos.    a  I       /-*  V 

^Y+Rsin.  a=0  5  -'-I       Y     ^      ^  r  '(A-) 
sin.  a 
Now  each  ofthese  equations  exists  sepam/c/y,  whatever  be  X,  orwhat- 
ever  be  Y,  because  R  is  always  equal,  and  opposite  to  the  pressure 

X  Y 

or—: > whatever  this  may  be;  and,  as  the  equilibrium 

cos.  a       sm.  a 

requires  that  they  exist  together,  it  is"  merely  necessary  that  we 

X  Y 

have =- or  X  tan.  a— Y=0  •  (B.) 

cos.o      sm.  a 

If  we  take  the  axes  of  components  the  one  parallel  and  the  other 
perpendicular  to  the  horizon,  then  the  angle  a  (fig.  22)  will  be  ob- 
tuse, and  COS.  a=  — cos  a'=  —  sin.  i  and  the  equation  of  condition 
just  deduced  is,  therefore,  in  this  case,  X  +  Y  tan.  i=0  •  (C) ;  i 
being  the  inclination  of  the  line  of  support  to  the  horizon.  It  must 
be  remembered  that  when  this  equation  is  satisfied,  and  we  wish  to 
determine  the  resistance  R  or  the  pressure  on  the  line,  we  must  recur 
to  one  of  the  equations  (A). 

From  what  has  now  been  said,  it  appears  that  the  equation  (C) 
expresses  the  conditions  of  the  equilibrium  of  a  heavy  body  upon 
any  curve  surface,  i  being  the  inclination  of  its  tendency  to  move, 


48  ELEMENTS  Of  STATICS. 

in  virtxie  of  its  weight,  to  the  horizon ;  and  the  horizontal  and  ver 
tical  axes  of  components  being  taken  in  the  vertical  plane  of  this 
tendency. 

We  shall  now  proceed  to  the  solution  of  a  few  problems. 

Problem  I. — (34.)  Given  the  inclination  i,  of  the  straight  line 
AC,  to  tlie  horizon  AB,  and  the  weight  of  a  heavy  body  W  to  de- 
termine what  weight,  P  acting  in  a  given  direction,  W  M  will  be 
sufficient  to  sustain  W  on  the  line  (fig.  23). 

Here  are  three  forces  acting  at  W,  viz.  the  weight  W  in  the  ver- 
tical direction  WY,  the  resistance  of  the  line  AC  acting  in  the  per- 
pendicular direction  WR,  and  the  weight  P  acting  in  the  direction 
WM,  and  all  these  directions  are  given  ;  hence,  as  one  of  the  forces 
W  is  given,  we  have  enough  to  determine  the  other  two. 

Let  us  call  tlie  given  angle  CWM,  (,  then  CWQ  being  equal  to 
90  +  1,  wc  have  MWQ=90+i  +  f,  which  call  0,  then  we  have  sin. 
MWQ=sin.  e,  sin.  MWR  =  cos.  t,  sin.  RWQ=sin.  QWN=  sin.  i, 
consequently,  calling  the  resistance  AVR,  R, 

_R sin.g  ^  sin.  e_      cos.  (i-fQ 

AV      COS.  ( '  '  COS.  f  cos.  ( 

P        sin.  i  sin.  i 


AV       COS.  s  cos,  t 

If  the  power  P  act  along  the  plane,  then  f=0,  and,  consequently, 

P      Sin*   i 
in  this  case,  R=W  cos.  {,  P=W  sin.  i  .-.  :=r= — '—.' 

K     cos.  I 

If  the  power  act  in  a  direction  parallel  to  the  horizon  then  t= — i: 

hence  R=AV :=AV  sec.  i, P=AV — ^.•.— =sin.  i. 

COS.  I  cos.  I      a 

If  the  power  act  in  a  direction  perpendicular  to  the  horizon,  then 

fi  =  180°,  also  COS.  f =sin.  i,  therefore,  R=0,  P=AV  ;  that  is,  there 

must  be  no  pressure  upon  the  line,  and,  therefore,  the  power  P  must 

be  equal  to  the  whole  weight  which  it  sustains.     If  we  suppose  the 

power  to  act  perpendicularly  to  the  line,  then  {=90°,  and  cos. 

(i-f-{)=sin.  I,  and  the  general  formulas  give 

sin.   i  sin.  i 

R  =  AV  — =  QD,P  =  AV — =  OD. 

These  equations  show  that  the  equilibrium  cannot  be  maintained 
under  these  circumstances,  unless  an  infinite  pressure  is  exerted 
on  the  line  requiring  an  infinite  power  P.  On  reviewing  the  fore- 
going results,  it  appears  that  the  power  P,  necessary  to  support  a 
weight  AV  on  an  inclined  plane,  will  be  the  least  possible  when  it 
acts  in  the  direction  of  this  plane  :  indeed  it  is  plain  from  the 


EQUILIBRIUM    ON    A   SURFACE.  49 

Sin.  t 
general  expression  P=W  — '—,  that  P  will  be  the  least  possible,  i 

remaining  the  same,  when  cos.  t==l. 

If  we  had  solved  this  problem  by  the  method  of  resolution, 
taking  for  axes  the  lines  WC,  WR,  as  recommended  at  the  former 
part  of  last  article,  then  the  single  equation  of  equilibrium  of  which 
we  have  there  spoken  would  have  been  P  cos.  f — W  sin.  i=0 .  (1) ; 
from  which  we  immediately  get  the  value  of  P  sought,  viz. 

p=w'^"''. 

cos.  £* 
To  determine  the  pressure  we  must  employ  the  equation  furnished 
by  the  other  component  forces,  viz.  those  acting  in  WR,  this  equa- 
tion is  R+P  sin.  t  —  W  cos.  i=0 (2.) 

„     „,  sin.  i  sin.s — cos.  i.  cos.  f 
or  R=W =0 

COS.  f 

„      „,  cos.  i  cos.  ( — sin.  i  sin.  t     „^  cos.  (  i  4-s) 
.'.  R=W =  W i^ <-. 

cos.  £  COS.  i 

By  employing  the  second  mode  of  resolving  the  forces,  that  is  ac- 
cording to  horizontal  and  vertical  axes,  we  should  have  for  the  con- 
ditions of  equilibrium  the  equation  (C),  in  this  case, 

P  cos.  (s+i)  -f-lP  sin.  (,+z_W?!^.=0. 
^  ^  cos.e 

whence  P  cos.  ?  cos.*^  i+P  cos.  i  sin.^  z=:  W  sin.  i 

.-.Pcos.  .=Wsin.i.-.P  =  W— -. 

COS.  s 

For  the  resistance  R  we  must  employ  one  of  the  equations  (A) ; 

1  •        1      /.  1         T^cos.  (s+i)     _,     ,,,  COS.  (f-fi)       ,    ^ 

taking  the  first  we  have  P r-^^—: — i=K=  W ^ ^  as  before. 

sm.  I  COS.  f 

Problem  II. — Given  the  position  of  the  line  AC,  and  of  the 
pulley  M,  as  also  the  weights  of  W  and  P,  to  determine  where- 
abouts AV  must  be  placed  that  the  equilibrium  may  be  possible 
(fig.  24). 

The  perpendicular  MM'  is  given  because  the  position  of  M  and 
of  AC  are  given.  Call  this  perpendicular  a;  then  by  equation  (1) ; 
last  proposition, 

W   .     .        .  ^P^—WHm.H 

cos.  a  =  — -sm.  I  .',  sm.  t= . 

P  p 

Now  WM  sin.  f=MM'  =  a 

...WM=    «  P'^ 


sin.f     sin.tVP^— W^sin.^i 
an  equation  which  determines  the  place  of  W. 
E  7 


50  ELEMENTS   OF    STATICS. 

Problem  III. — Two  weights  W,  W,  attached  to  the  extremi- 
ties of  a  siring,  which  passes  over  a  fixed  puUy,  mutually  support 
each  other  on  two  inclined  planes  (fig.  25),  to  determine  the  rela- 
tions between  W,  W',  the  tension  of  tlic  string,  and  the  pressures 
on  the  planes. 

Here  each  weight  is  supported  by  the  tension  of  the  string,  which 
is  the  same  throughout ;  this  tension  will  then  be  the  same,  as  re- 
gards each  body,  as  tlie  power  we  have  hitherto  called  P. 

If,  therefore,  we  designate  the  angles  concerned  in  one  of  the 

planes  as  m  prob.  I.,  we  have  r^= ^ 1 

VV  COS.  t 

In  like  manner,  for  the  other  plane  we  have 

R'     cos.  (i'  +  f')     P       sin.  i'    _   W  _sin.i' cos.  r  _ 

W  cos.  t'     '   W     cos.  t' '  *  W'     sin.icos.f'' 

which  equations  exhibit  the  relations  required. 

If  the  pulley  be  fixed  at  tlie  intersection  of  the  planes,  so  that 

the  string  acts  in  each  plane,  then  f — 0,  t' — 0  and 

W      sin.  i'      CA      ,      .      ,  .  ,  •     ,  •  i    i- 

——=— — :-=-—--;  that  IS,  the  weights  are  m  this  case  as  the  lines 
W      sin.  i      CA'  ^ 

on  which  they  rest. 

It  has  been  already  observed  that  when  the  weight  rests  on  a 
point  of  a  curve,  the  conditions  are  the  same  as  if  it  rested  on  the 
tangent  line  through  that  point;  the  inclination  of  this  line  to  the 
horizon,  which  it  is  necessary  to  know,  may  be  determined  when 
we  know  the  equation  of  the  curve,  referred  to  vertical  and  hori- 
zontal axes,  and  the  co-ordinates  of  the  point  where  tlie  body  is 
placed.  If  we  resolve  all  the  forces  whicli  are  applied  to  the  point, 
in  the  directions  of  the  axes,  and  call  the  inclination  of  the  tangent 
line  to  the  axis  of  x,  i,  then  we  know  that  the  conditions  of  equi- 
librium will  be  expressed  by  the  single  equation  (C),  at  page  47, 
viz.  X  +  Y  tan.  i=0;  but,  if  x,  y  are  the  co-ordinates  of  the  point, 

we  know  (jDiff".  Calc.  p.  113,)  that  tan.  i—-r'i  hence  the  equation 

^  ux 

of  condition  is  X-fY  -^=0  ....  (1.) 
dx 

If  a  weight  W  be  supported  on  a  curve  by  means  of  anotlier 
weight  P  (fig.  26),  hanging  vertically,  the  two  weights  being  con- 
nected by  a  flexible  string  passing  over  a  pulley,  then  it  will  be 
most  convenient  to  take  the  vertical  line  MX  as  axis  of  x,  and  the 
horizontal  line  MY  as  axis  of  y.  In  this  case  we  shall  have  the 
following  values  for  the  forces  X  and  Y,  viz.  for  X  we  shall  have 
the  weight  W  diminished  by  P  cos.  MWn=P  cos.  WM7/1,  and 
for  Y  we  shall  have — P  cos.  MWw,  that  is,  putting  MW=r, 


EQUILIBRIUM  ON  A  SURFACE.  51 

X=W— P-,  Y=— P^; 
r  r 

substituting  these  values  in  the  above  equation,  we  have 

T         r     ax  r         "^  ax 

hni,s\ncer^=x^-{-y^.-.r -Y-^x-\-y-~-;  hence  the  equation  ol  equi- 

Ubrium  is  W-P-^=0 (2). 

Problem  IV. — A  given  weight  W  rests  upon  a  circular  arc,  as 
in  fig.  26,  being  supported  by  another  given  weight  P,  by  means 
of  a  string  passing  over  a  pulley,  fixed  at  a  given  point  in  the  ver- 
tical line  MX,  passing  through  the  centre  C :  to  determine  the  po- 
sition of  W. 

Referring  the  curve  to  the  vertical  and  horizontal  axes  MX,  MY, 
and  calling  MC,  x',  we  have,  for  the  equation  of  the  curve,  x^ — ■ 
2x'x-\-x^^y^=:r" ;  for  substituting  /,  the  length  of  the  string  MW, 
for  x^-\-y",  I-  —  2x'x=r^  —  x'^.  Hence  differentiating  with  respect 

dl  dl        x' 

to  x,l  — x'=0.-. —j—=-j- ,  so  that  the   general  equation  of 

equilibrium  (2)  is,  in  this  case,  , 

W-P— -0-/-— • 

which  gives  the  position  of  W.     Or,  because 

pa  ^'        y3 

l^=x--\-y''=r^A-1x'x — a:'^  .•.a^=Mm={-^^^+l }— — -—p. 

Problem  V. — Instead  of  a  circle  let  the  curve  of  support  be  an 
hyperbola  with  its  transverse  diameter  vertical,  the  pulley  being  in 
the  centre  (fig.  27). 

The  equation  of  the  curve  is  a^  y"^ — b^  x^= — a^  b^ ;  and  the 
expression  for  /,  the  distance  of  any  point  in  it  from  the  centre,  is 
{Anal.  Geom.  p.  143-4,) 

a^-i-b^    ■  a^4-b^ 

l^= — - —  x^  —  63_g2  ^2  —  53 .  g2  being  put  for  — ^— . 
a^  ^  ^  a^ 

Hence,  by  differentiating, /-y- =6^3:  .*.  — — =—j—, 

and  the  equation  of  equilibrium  is,  therefore, 

W_P^==0,...-f=cos.AM=^,. 


G2  ELEMENTS  OF  STATICS. 

This  fixes  the  position  of  W ;  or  if  we  substitute  for  /  its  value  in 
terms  of  e  and  x,  as  given  above,  we  shall  have 

X^  =  ^ X"—  TT .      .-.  x=  — —  =  M7». 

It  appears  from  this  expression  that  the  equilil)rium  is  impossible 
if  W  is  less  than  Pe ;  and  if  W=Pe  the  point  of  rest  must  be  at 
an  infinite  distance. 

Problem  VI. — It  is  required  to  find  a  curve  such  that  a  given 
weight  P  hanging  over  the  pulley  may  balance  another  given  weight 
W  at  every  point  of  it  (fig.  28). 

We  have  here  to  find  a  curve  such  that  the  equation 

W-P^=0,„rW-P'^!+^=O 
ax  ax 

may  exist  not  only  at  one  particular  point,  as  in  the  preceding  cases, 

but  at  every  point  (x,  y)  of  the  curve.      This  equation,  therefore, 

can  be  no  other  than  the  diff'erential  equation  of  the  sought  curve. 

Hence,  multiplying  by  dx  and  integrating,  there  results 

Wa;  — Pr+C=0,  or  Wa:—  P  ^/x^+  y-|-C=0,  or 

pa_W2     ,       2WC         C^ 

y'+  — p7—  ^'  — pi-^'-- pr=o  •  •  •  •  (1); 

for  the  equation  of  the  curve  sought.  In  order  to  simplify  this 
equation,  let  us  remove  the  term  containing  the  first  power  of  x, 
which  is  done  by  substituting  for  x,  in  this  equation,  the  value  (See 

WC 
Anal.  Geom. p.  173-4, )a:=.p^ wi""^-^'  which  leads  to  the  equa- 

tion,  Y^ pj X2= (2);  and  this  equation 

characterizes  an  hyperbola  related  to  its  principal  axes.  For  the 
distance  c  between  the  centre  and  focus  of  this  hyperbola,  we  have 

WC 

{Anal.  Geom.,  p.  171,)  c=^^rp; p^;  but  this  is  the  distance  of 

the  new  origin  from  the  primitive  origin,  and  the  primitive  origin 
IS  on  the  pulley ;  hence  the  pulley  is  at  the  focus  of  the  hyperbola. 
By  putting  first  Y=0  and  then  X=0  in  the  equation  (2),  we 
have  for  the  semi-axes  of  the  hyperbola, 

>A  P^         B  ^ 

'*^— W»— P«'         (W— P»)t    ' 

in  which  equations  C  is  arbitrary. 


PARAtLEL  FORCES.  *  53 

SECTION  II. 

ON  THE  EQUILIBRIUM  OF  A  SOLID  BODY. 

(35.)  Having  considered  pretty  much  at  large  the  equilibrium  of 
forces,  acting  upon  a  free  point,  it  is  time  now  to  examine  the  more 
general  case  in  which  forces  act  upon  different  points,  all  connected 
together  in  an  invariable  manner,  as  we  shall  here  suppose  the  parts 
of  a  solid  body  to  be.  We  shall  divide  the  theory  into  two  parts  ; 
first,  considering  the  forces  which  act  upon  the  body  to  be  all 
parallel,  and  then  considering  them  to  act  in  any  manner  whatever. 


CHAPTER  I. 

ON  PARALLEL  FORCES 

(36.)  Let  us  first  consider  two  parallel  forces  P,  P,,  acting  at  the 
extremities  of  a  straight  line  AB,  (fig.  29,)  and  let  it  be  required  to 
determine  what  must  be  the  intensity,  and  where  the  point  of  ap- 
plication, of  a  single  force,  which,  acting  on  the  line,  shall  have  the 
same  effect  as  these  two.  Let  us  represent  the  parallel  forces  by 
the  lines  AP,  BPj,  and  let  us  apply  to  the  extremities  of  the  line 
any  two  equal  but  opposite  forces  AM,  BMj  ;  these  will  destroy 
each  other,  and  will,  therefore,  have  no  effect  on  the  system. 
Hence,  instead  of  the  two  forces  AP,  BPj,  acting  on  the  line,  we 
may  consider  as  acting  the  four  forces  AM,  AP,  BMj,BPj,  or  the 
resultants  of  these,  AR,  BRj.  We  have  thus  exchanged  our  two 
parallel  forces  for  two  oblique  forces,  meeting  in  some  point  C. 
Considering  the  lines  to  be  all  rigid,  we  may  transfer  the  points  of 
application  of  these  forces  to  their  point  of  concurrence  C,  making 
CE=AR,  and  CEi=BRj,  so  that  these  concurring  forces,  acting 
on  C,  have  the  same  effect  on  the  rigid  line  AB,  with  which  they 
are  connected  by  the  rigid  lines  CA,  CB,  as  the  original  forces  P, 
Pj.  Let  us  now  resolve  the  forces  CE,  CEj,  into  their  original 
components  Cm, C^, and  C?n,,Cj9j  ;  then  since  the  two  Cm, Cm^, 
are  equal  and  opposite  they  destroy  each  other,  so  that  the  system 
will  be  reduced  to  the  two  conspiring  forces  C;;,  Cp^,  or  to  the 
single  force  C/j  +  Cp^,  which  is  equal  to  AP  +  BPj ;  thus  we  have, 
for  the  intensity  of  the  resultant  of  the  two  forces  P,  P^,  P+Pj  = 
R ;  and  as  this  force  may  be  applied  at  any  point,  in  its  direction 
e3 


M  ELEMENTS    OF    STATICS. 

CO,  0  will  bp  that  point  in  the  proposed  line  to  which  it  must  be 
applied  ;  the  situation  of  this  point  is  thus  determined.  By  similar 
triangles, 

C0_^  OBE/?i        0B_  Cp.Ep,_  Cp 

A0~1e^  ^"     CO  "(1^1  '"■  AO~  Ep.i:p,~Cp^  ' 
that  is  OB  :  AO  : :  Cp  :  Cp^,  or  Cp,  Cp^  being  equal  to  P,  Pj  ; 
OB  :  AO::  P:P,. 

Hence  we  conclude  that  the  resultant  of  two  parallel  forces  is 
also  parallel,  is  equal  to  their  sum,  and  acts  at  that  point  which  di- 
vides the  distance  between  them  into  parts  reciprocally  proportional 
to  their  intensities.  This  point  therefore  is  fixed  however  the  di- 
recticn  of  the  components  may  vary. 

The  proportion,  just  deduced,  gives  also  (see  fig.  30,) 
AB:  OB::  R  :  P 
AB:AO::R:Pi; 
therefore  P,  Pj,  and  R  are  to  each  other,  respectively,  as  OB,  Oa, 
AB,  that  is  to  say,  that  any  two  of  the  three  forces  are  to  each  other 
reciprocally  as  their  distances  from  the  third,  so  that  when  any 
three  of  the  six  quantities  concerned,  viz.  the  three  forces  and  the 
three  distances  are  given,  the  other  three  may  be  determined  by  the 
successive  application  of  this  theorem.  The  same  theorem  then 
serves  to  divide  a  given  force  R  into  tAvo  others  parallel  to  it,  acting 
at  given  distances  OA,  OB,  on  each  side  of  0.  And  lastly,  it  serves 
also  to  determine  the  intensity  and  point  of  application  of  that  force 
Pj  (fig.  31)  which  will  keep  in  equilibrium  t!ie  line  AO,  acted  upon 
by  two  opposing  parallel  forces  P,  R',  whenever  such  equilibrium 
is  possible.  This  qualification  is  necessary,  because  there  is  one 
case  in  which  two  parallel  forces,  acting  on  opposite  sides  of  a  line, 
cannot  be  equilibrated  by  any  third  force,  viz.  the  case  in  which  the 
iwo  forces  are  equal;  for  it  is  plain  that  if  R'  be  equal  to  P,  that  R, 
which  is  equal  and  opposite  to  R',  cannot  be  the  resultant  of  P,  and 
any  other  force  Pj,  because  if  it  were  we  should  have  P  +  Pj  =R, 
whereas  P  alone  is  equal  to  R  :  hence  Pj  must  be  0,  and  therefore, 
by  the  theorem,  tlie  point  of  application  B  must  be  infinitely  distant, 
so  that  no  single  force  can  keep  the  line  AO  at  rest  when  its  extremi- 
ties are  solicited  by  equal  parallel  forces  acting  in  opposite  directions. 
The  tendency  of  these  forces  will  plainly  be  to  cause  the  line  to  turn 
about  its  middle  point,  this  being  at  rest. 

From  what  has  now  been  said  it  follows  that  the  resultant  of  two 
opposite  forces,  P,  R',  applied  to  different  points.  A,  O,  is  equal  to 
their  difference  P/,  acting  parallel  to  them  in  the  direction  oj  the 
greater,  and  that  its  point  of  application  B  is  given  by  the  pro- 
portion. 

P',  :  R'  ::  AO  :  AB  .•.AB=^;1a0=    p,  ^  p  AO. 

r  1  K  —  " 


PARALLEL    FORCES.  55 

From  knowing  how  to  compound  two  parallel  forces  acting  upon 
a  straight  line,  we  are  enabled  to  compound  any  number  so  acting. 
The  resultant  will,  obviously,  be  equal  to  the  algebraic  sum  of  the 
components,  affixing  opposite  signs  to  those  which  draw  in  opposite 
directions.  As  to  the  point  of  application  of  this  resultant  we  shall 
not  stop  to  determine  it  for  this  particular  case,  but  shall  proceed  to 
consider  the  theory  of  parallel  forces  in  all  its  generality. 

(37.)  Let  P,  Pj,  Pj,  &c.  be  parallel  forces,  applied  to  any  sys- 
tem of  points.  A,  Aj,  Aj?  &c.  any  how  situated  in  space,  but  inva- 
riably connected  by  rigid  lines,  (fig.  32,)  and  let  it  be  required  to 
determine  the  resultant  of  this  system  both  in  intensity  and  position. 

The  most  obvious  mode  of  proceeding  is  this,  viz.  first  to  com- 
pound two  of  the  forces  P,  P^,  and  to  substitute  for  them  their  re- 
sultant R;  then  to  compound  this  with  the  third  force  Pg,  and  to 
substitute  for  the  two  R,  P^,  that  is,  for  the  three  P,  P^,  P^,  their 
resultant  Rj,  and  so  on  till  the  system  is  reduced  to  the  two  paral- 
lel forces  R„ — 1,  P„,  of  which  the  resultant  will  be  that  of  the 
whole  system  ;  and  will,  therefore,  be  equal  in  intensity  to  the 
sum  of  the  components.  In  this  process  of  composition  the  several 
partial  resultants  R,  Rj,  Rg,  &c.  are  not  only  determined  in  inten- 
sity, but  the  point  of  application  in  the  line  joining  the  points,  acted 
on  by  the  two  components,  is  in  each  case  determined.  Every  such 
point  would  remain  fixed,  however  the  direction  of  the  component 
parallel  forces  might  vary,  provided  their  respective  intensities  did 
not  vary  (36) ;  and,  therefore,  the  point  of  application  of  the  final 
resultant  would  remain  fixed,  however  the  direction  of  the  system 
of  parallel  forces  might  vary  provided  they  retained  their  respective 
intensities  ;  it  is  through  this  point,  therefore,  that  the  resultant  of 
the  system  must  always  pass  under  every  change  of  direction  ;  it  is 
hence  called  the  centre  of  these  parallel  forces. 

(38.)  Let  now  there  be  assumed  any  three  rectangular  axes,  and 
let  us  represent  by 

X,  y,  z,  the  co-ordinates  of  the  point  A 
•^i»  ?/if  ^i»      •         •         •         •         Aj, 

•^2 '2/2'  ^2'  •  •  •  •  A.2» 

&c.  &;c. 

and  by  X,  Y,  Z,  those  of  the  centre  of  tlie  parallel  forces  ;  we  shall 
prove  that  these  latter  are  severally 

-        p  +  p^+p^... 

y_Py+P,yr+P,y,  . 
P+P.+P^... 

^-     P4-Pi+P,  ....    J 


^...(1). 


56  ELEMENTS    OF    STATICS. 

For  let  us  first  consider  only  two  forces  P,  Pj,  acting  on  the  points 
A,  A,  (fiff.  33),  of  which  the  abscissas  are  OA'=x  AO,  '=x\,  and 
let  C  be  the  centre  of  these  two  forces,  its  abscissa  being  OC '  '=X' ; 
then  if  ff,  c,  Oj,  be  the  projections  of  A,  C,  A^,  on  the  plane  of  xy, 
we  shall  have,  on  account  of  the  parallels, 

A,C  :  AC  ::  a^c  :  ac  ::A.\C=x^ — X'  :  A'C=X' — x; 
but  (36),  A,C  :  AC  ::  P  :  P,  .-.  x,— X'  :  X'—x  ::  P  :  Pj 

.•.(P+PJX'=Px+P,x,  ...X'  =  -^-^^. 

Let  us  now  proceed  with  the  two  forces  (P+Pj)  and  P2,  after 
having  joined  their  points  of  application  C,  A,,  exactly  as  we  have 
proceeded  with  P  and  Pj,  and,  calling  the  abscissa  of  the  centre  of 
our  new  forces  X",  the  result  must  be 

(P+P.+P,)X"=(P+PJX'+P,x,  ; 
or,  substituting  for  X'  the  value  just  obtained 

(P+P.-f-P^)  X"=Vx-^V,x,+F^x^ 
Px-\-V,x,-^P^x^ 

'    "^  -     P  +  P1+P3        • 

Proceeding  in  this  manner  till  we  arrive  at  the  centre  of  all  the 
parallel  forces,  of  which  the  abscissa  is  X,  we  shall  have,  finally, 
Fx-^V,x,-\-P,X2  ... 
P+P1+P2... 
as  announced ;  and   if  for  the  axis  of  x  we  substitute  successively 
the  axes  of  y  and  of  z,  we  shall  have  the  similar  equations 

V    Py+Piyi+P^y^.-. 
^-      p+p^+p^... 

Pz  +  PlZ,+P2~2    •    ■   • 
P  +  P,+P,    ...  ' 

and  thus  we  may  always  determine  the  co-ordinates  of  the  centre 

when  we  know  those  of  the  points  of  application  of  the  system  of 

parallel   forces,  as  well  as  the  several  intensities  of  those  forces. 

Calling  the  resultant  of  the  forces  R,  the  preceding  equations  give 

RX=Px+Pi  x.+P^x^  '  ") 

RY=P^  +  P,3/,+P,3/,  .    1.(2). 

RZ=Pr+P,  r,+P,^3  .  J 

We  may  here  remark  that  it  is  possible  so  to  place  the  axes  of 

co-ordinates,  that  two  of  the  three  equations  (2)  will  suffice  to  fix 

the  position  of  the  resultant  of  the  system ;  for  let  one  of  the  axes, 

as  the  axis  of  z,  be  taken  parallel  to  the  direction  of  the  forces, 

then,  as  the  resultant  itself  will  be  parallel  to  the  same  axis,  its 

position  will  be  known  if  we  only  know  where  it  meets  the  plane 

of  xy,  that  is,  if  we  know  the  X,  Y,  of  any  point  in  it ;  hence  the 

two  first  of  equations  (2)  are  sufficient  to  determine  the  line  in 

which  the  resultant  acts,  and  this  is  all  we  want  to  know,  since  on 


PARALLEL  FORCES.  57 


whatever  point  in  this  line  it  acts,  the  effect  is  the  same.  Under  this 
arrangement  of  the  axes,  therefore,  the  equations  necessary  for  the 
determination  of  the  resultant  in  intensity  and  position  are 

R=P+P,+P2  +  P3+  •  ■) 

RX=Pa^+P,  a^.+P^  X2  +  P3  x,+  •  y  •  ■  (3). 
RY=Pi/+P,  2/1 +P3  2/2+P3  2/3+  •  J 
(38.)  The  product  of  any  force,  by  the  perpendicular  distance  of 
the  point  on  which  it  acts  from  any  plane,  is  called  the  moment  of 
that  force  with  respect  to  the  plane  ;  thus  Fx  is  the  moment  of  the 
force  P  with  respect  to  the  plane  of  YZ,  because  x  is  the  distance 
of  the  point  A  on  which  it  acts  from  that  plane.  Hence  we  learn 
from  either  of  the  three  equations  (2)  just  given,  that  the  moment 
of  the  resultant  of  a  system  of  parallel  forces  in  reference  to  any 
plane  is  equal  to  the  sum  of  the  moments  of  the  components  in 
reference  to  the  same  plane ;  the  algebraical  sum  being  always  un- 
derstood, regard  being  had  to  the  signs  of  the  forces  as  well  as  to  the 
co-ordinates  of  the  points  on  Avhich  they  act.  It  is  easy  to  see  how 
the  foregoing  results  become  abridged  when  the  forces  all  act  in  one 
plane,  as  also  when  the  several  points  on  which  they  act  are  in  one 
straight  line  ;  in  the  former  case  only  one  co-ordinate  plane  is  ne- 
cessary, viz.  the  plane  in  which  all  the  points  are  situated ;  in  the 
latter  case  only  one  axis  is  necessary,  viz.  the  line  in  which  the 
points  are  situated,  so  that  either  one  or  two  of  the  foregoing  gene- 
ral equations  may  in  particular  cases  become  superfluous. 

The  preceding  theory  will  enable  us  readily  to  determine  the  con- 
ditions of  equilibrium  of  a  system  of  parallel  forces ;  for  let  us  as- 
sume the  axis  of  z  parallel  to  the  direction  of  the  forces,  then,  since 
the  sum  of  the  forces,  that  is  the  resultant  R,  is  0,  we  have,  by  the 
equations  marked  (3), 

P  +  P,+P,-f  P3-f  .  =0 
Px+P^x,-{-F,x,+F,x,+  .  =0  [-  .  (4). 
Py+P,y,+P,2/,+P3  2/3+ 
which  are  the  equations  necessary  to  establish  the  equilibrium,  and 
they  express  these  conditions,  viz. 

1st.   The  sum  of  the  forces  must  be  equal  to  0. 
2d.   TTie  sum  of  their  moments,  in  reference  to  each  of  two  per- 
pendicular jAanes  parallel  to  their  direction,  must  be  equal  to  0. 

(39.)  Before  terminating  this  chapter,  we  should  remark,  that  a 
more  concise  notation  is  frequently  employed  to  express  the  equa- 
tions (1),  (2),  &c.  thus  the  equations  (1)  are  written 
Y      S  (Pa;)  ^     s  (Pv)  „      S  (Vz)    ,        ,  .... 

sTpT  sTpT  SfTT'         character    S  signifymg 

the  sum  of  the  whole  system  of  quantities  of  the  form  of  that  to 
which  it  is  prefixed.     In  like  manner  the  equations  (2)  may  be 

8 


'::} 


58  ELEMENTS  OF  STATICS. 

written  RX  =  S  (Pa-),  RY=s  (Py),  RZ=r  (Tz), 

and  the  equations  of  equilibrium  according  to  this  notation  are 

2  (P)-0,  2  (Px)=0,  S  (Pt/)=0, 
provided  the  two  perpendicular  planes  to  which  the  moments  are  re- 
ferred are  parallel  to  the  direction  of  the  forces.  We  shall  now  pro- 
ceed to  some  interesting  and  important  applications  of  the  theory  de- 
livered in  this  chapter. 


CHAPTER  II. 

ON  THE  CENTRE  OF   GRAVITY. 

(40.)  Experience  teaches  us  that  all  bodies  within  our  reach  tend 
towards  the  earth,  to  which,  if  abandoned  to  themselves,  or  left  un- 
supported, they  would  fall  in  a  vertical  direction.  The  reason  why 
smoke  and  vapours  in  general  do  not  fall  to  the  earth,  is  that  they 
are  not  left  unsupported,  being  indeed  borne  up  by  the  air  in  the 
same  way  that  a  piece  of  wood  is  borne  up  by  the  water  in  a  vessel, 
and  prevented  from  reaching  the  bottom  as  it  would  do  if  this  sup- 
port were  removed.  This  universal  tendency  of  all  bodies  to  the 
earth,  proves  the  existence  of  a  soliciting  power  whose  influence 
extends  equally  to  every  body  with  which  we  are  surrounded.  By 
the  tendency  here  spoken  of  we  mean  the  disposition  to  move,  and 
that  this  is  the  same  in  all  bodies,  great  and  small,  when  all  support 
is  taken  away,  has  been  fully  and  frequently  established  by  the  most 
convincing  experiments  ;  thus  if  a  very  small  particle  be  placed  be- 
side a  large  mass  in  a  vessel  exhausted  of  air,  they  will,  when  let 
go,  continue  beside  each  other  during  the  whole  time  of  descent,  and 
will  both  strike  the  bottom  of  the  vessel  at  the  same  instant,  so  that 
if  it  were  possible  to  destroy  the  cohesion  among  the  particles  of 
matter,  in  virtue  of  which  it  becomes  a  solid  mass,  thus  enabling 
each  particle  to  obey  whatever  force  acted  upon  it  individually,  yet 
the  tendency  to  move  being  exactly  the  same  in  each,  no  one  parti- 
cle could  in  descending  displace  any  other,  so  that  the  same  ar- 
rangement would  be  preserved  during  the  descent  as  if  all  the  par- 
ticles cohered.  We  call  this  soliciting  power  of  the  earth,  which 
we  see  is  altogether  independent  of  the  mass  on  which  it  acts,  the 
FORCE  OF  GRAVITY,  or  simply  gravity,  thus  naming  an  influence,  the 
nature  of  which  we  know  nothing  only  as  regards  its  effects,  and 
this  is  in  fact  all  that  we  here  require  to  know  of  it. 

It  may  be  proper  here  to  caution  the  student  against  a  ver}'  com- 
mon misapplication  of  the  term  gravity.    We  use  incorrect  language 


CENTRE  OF  GRAVITY.  59 

when  we  speak  of  the  gravity  of  this  body,  or  the  ^avity  of  that,  for 
gravity  is  not  of  the  body  but  of  the  earth,  and  is  always  the  same 
at  the  same  place,  exerting  the  same  effect  on  all  bodies  however  dii- 
ferent,  that  is,  producing  in  all  the  same  tendency  to  move.  The 
weight  of  a  body  furnishes  us  with  no  information  respecting  the 
force  of  gravity,  but  only  with  respect  to  the  number  of  its  constituent 
particles,  for  if  one  body  is  double  the  weight  of  another  this  does 
not  arise  from  any  variation  in  the  force  of  gravity,  but  because  there 
are  twice  the  number  of  particles  in  one  body  that  there  are  in  the 
other,  and  each  particle  is  influenced  alike  ;  so  that  it  will  require 
double  the  effort  to  support  one  that  it  requires  to  support  the 
other ;  the  lighter  body  may,  however,  have  more  external  surface 
or  appear  under  gi-eater  bulk  than  the  heavier,  but  then  the  pores 
which  separate  the  component  particles  M-ill  be  proportionally  larger. 

Understanding  by  the  weight  of  a  body  the  effort  necessary  to 
prevent  its  falling,  we  may  correctly  say  that  the  weight  of  a  body 
is  the  resultant  of  all  the  efforts  (or  weights)  which  gravity  im- 
presses upon  its  component  particles ;  as  these  component  efforts 
are  all  directed  in  parallel  lines,  their  resultant  must  be  equal  to  their 
sum,  and  act  in  their  common  direction  and  at  a  point  which  will  be 
the  centre  of  these  parallel  forces,  and  which  in  the  present  case  is 
called  the  centre  of  gravity  of  the  body.  The  determination  of  this 
centre  in  different  bodies  may  be  effected  by  the  application  of  the 
theory  delivered  in  the  preceding  chapter,  provided  we  suppose,  as 
we  shall  here  do,  that  the  bodies  proposed  are  perfectly  homoge- 
neous, so  that  the  effort  necessary  to  counterbalance  the  influence  of 
gravity  on  any  part  of  the  body  will  be  proportional  to  the  mass  of 
that  part. 

Before  proceeding  to  particular  applications  of  the  theory,  we  may 
as  well  here  notice  the  distinguishing  characteristics  of  the  point 
which  we  have  called  the  centre  of  gravity,  and  which  are  direct  in- 
ferences from  that  theory  ;  these  are,  1st,  that  if  the  centre  of  gra- 
vity be  supported,  the  whole  body  will  be  in  equilibrium,  because 
the  resultant  of  all  the  forces  which  act  on  it  will  be  opposed  in 
whatever  position  the  body  be  plaped :  moreover  every  body  kept 
in  equilibrium  by  a  single  force,  must  have  its  centre  of  gravity 
in  the  line  of  direction  of  that  force.  2d.  The  sum  of  the  pro- 
ducts of  each  particle  of  a  body  into  its  distance  from  any  plane, 
the  distances  on  opposite  sides  taking  opposite  signs,  is  equal 
to  the  product  of  the  whole  mass  into  the  distance  of  its  cen 
tre  of  gravity  from  the  same  plane ;  so  that  if  a  plane  divide  a 
body  into  symmetrical  halves,  it  must  pass  through  the  centre  of 
gravity.  The  first  of  these  properties  points  out  an  experimental 
method  of  finding  the  centre  of  gravity  of  a  body :  thus,  let 
the  body  be  suspended  by  a  string  attached  to  any  point,  it  will 


60  ELEMENTS    OF    STATICS. 

arrange  itself  so  that  this  string  would,  if  we  could  continue  it,  pass 
through  the  centre  of  gravity.  In  like  manner,  if  it  were  suspended 
from  any  other  point,  the  line  of  the  siring  would  also  pass  through 
the  centre  of  gravity,  consequently  the  intersection  of  these  two 
lines  would  determine  that  centre.  If  the  body  have  a  Hat  surface, 
we  may  lav  it  on  a  horizontal  table  pushing  it  more  and  more  over 
the  edge  till  it  just  balances  itself,  in  which  position  the  centre  of 
gravity  will  be  vertically  over  the  edge  of  the  table ;  if,  then,  we 
mark  the  line  of  the  edge  on  the  body,  and  proceed  in  the  same  way 
with  the  body  in  another  position,  we  shall  thus  have  a  point  in  the 
same  vertical  as  the  centre  of  gravity,  and  to  which  if,  as  a  support, 
an  indefinitely  slender  vertical  rod  were  applied,  and  the  table  re- 
moved, tlie  body  would  remain  in  equilibrium.  Or  if  it  were  to  be 
suspended  by  this  point,  the  flat  surface  would  assume  a  horizontal 
position- 

(41.)  We  shall  now  investigate  general  analytical  expressions 
for  the  determination  of  the  centre  of  gravity  of  any  body  whatever. 

Let  ABC,  Sic.  [fig.  34,)  represent  any  solid  body,  the  component 
particles  of  which  we  shall  call  P,  P,,  V^^Sic.  and  their  sum  or 
the  mass  of  the  whole  body,  B.  Then,  if  G  be  the  centre  of  gra- 
vity of  this  body,  and  the  body  be  referred  to  three  rectangular 
planes,  the  distance  of  G  from  the  plane  of  zy  will,  by  equation  (1), 

page  55,  be  X=GH=^'''^^^'^^^^^'^^"^  "  '  ' (1). 

The  numerator  of  this  fraction  consists  of  the  sum  of  the  in- 
numerable particles  P,  P^,  P2,&c.,  multiplied  by  their  respective 
distances  from  the  plane  of  zy;  but  although  the  terms  are  innu- 
merable, yet  their  sum  may  be  accurately  determined  by  the  aid 
of  the  integral  calculus.  In  order  to  this  determination,  let  CN  be 
any  increment  of  the  body,  then  the  corresponding  increment  of 
the  expression  under  consideration,  that  is  of  the  numerator  of  (1), 
will  be  equal  to  the  sum  of  all  the  particles  in  the  slice  CN,  mul- 
tiplied by  their  respective  distances  from  the  plane  of  zy.  Now, 
calling  the  increment  MN  of  the  abscissa,  //,  it  is  obvious  that 
however  small  we  take //,  that' is  how^ever  slender  the  slice  CN 
may  be,  the  sum  of  which  we  have  just  spoken  will  always  be 
comprised  between  these  two,  viz.  the  sum  of  the  same  particles 
when  multiplied  each  by  the  distance  AM=.2',  and  the  sum  when 
multiplied  each  by  AN=ar+A ;  that  is,  putting  S  for  the  expres- 
sion we  are  considering,  and  A  S,  A  B,  for  the  corresponding  incre- 
ments of  this  and  of  the  body,  A  S  will  always  be  intermediate 
between  .r  A  B  and  (.r-f /<)  aB,  but  the  ratio  of  these  is 

(.t+ZOaB      ,  .     ^    ,.    . 

ii — =  1  m  the  limit, 

xaB 


CENTRE    OF    GRAVITY.  61 

or  when  h,  and  consequently  A  B,  is  0;  therefore  the  ratio  of  the 
intermediate   quantity  A  S  to  either  must  in  the  limit  be  1  ;  that  is, 

^—=1  in  the  limit,  that  is,  -^=].-,f/S=:rrfB  .-.  S=/j;rfB  ; 
hence  the  expression  (1)  is 

X=GH  =  -/i^] 

B     I 

In  like  manner  S.  .  .  .  .  (A), 

fyclB 


Y=-^ 


B 


equations  from  which  the  co-ordinates  of  the  centre  of  gravity  of 
B  may  be  determined  when  the  equation  of  B  is  known. 

But  it  must  be  observed,  that  though  in  all  these  equations  rfB 
signifies  the  differential  of  the  body,  yet  it  is  not  to  be  represented 
in  all  by  the  same  analytical  expression  :  for  regard  must  be  had 
to  the  position  of  the  slice  A  B,  as  this  will  in  general  be  different 
in  its  three  positions,  parallel  to  the  rectangular  planes  ;  and  there- 
fore also,  in  general,  the  expressions  for  dB  will  all  three  be  dif- 
ferent; but  this  will  be  shown  more  clearly  when  we  come  to 
apply  the  formulas  to  the  determination  of  the  centres  of  gravity 
in  surfaces  and  solids,  (art.  43.) 

It  is  seldom,  however,  requisite  to  employ  all  three  of  these 
equations  for  that  purpose ;  much  will  depend  upon  a  happy  ar- 
rangement of  the  co-ordinate  axes :  thus,  if  we  know  the  position 
of  a  plane  that  will  divide  the  body  into  symmetrical  halves,  then 
we  know  that  the  centre  of  gravity  must  lie  in  this  plane,  (p.  59 ;) 
taking,  therefore,  this  for  one  of  the  co-ordinate  planes,  it  is  clear 
that  two  of  the  equations  (A)  will  suffice  to  determine  the  centre. 
If  we  know  two  perpendicular  planes,  of  Avhich  each  divides  the 
body  into  symmetrical  halves,  and  in  all  bodies  of  revolution  any 
two  planes  through  the  axis  of  revolution  will  do  this,  then  we 
also  know  the  line  in  Avhich  the  centre  lies,  and  hence  one  of  the 
above  equations  will  be  sufficient.  Should  the  body  be  merely  a 
lamina  of  matter,  so  thin,  indeed,  as  to  be  taken  for  a  plane  sur- 
face, then,  by  choosing  the  axis  in  this  plane,  more  than  two  of 
the  foregoing  equations  can  never  be  requisite,  and  but  one  if  one 
of  the  co-ordinate  axes  divide  the  figure  symmetrically  into  halves ; 
and  the  same  is  obviously  true  if  the  body  be  considered  merely  as 
a  plane  line.  If  the  curve  be  of  double  curvature,  all  three  of  the 
equations  will  generally  be  necessary. 

Let  us  now  proceed  to  the  actual  determination  of  the  centres  of 
gravity  in  given  figures,  considering  in  order  lines,  surfaces,  and 
solid  bodies. 
F 


62  ELEMENTS  OF  STATICS. 

Determination  of  the  Centre  of  Gravity  of  a  Plane  Line. 

(42.)  When  the  body  may  be  considered  as  a  line  lying  in  one 
plane,  we  shall  put  2  «  for  B  ;  and  supposing  first  that  the  line  is 
symmetrically  situated  with  respect  to  the  axis  of  x,  that  is,  that  the 
centre  is  in  this  axis,  we  shall  have  by  the  first  of  (A)  this  expres- 
sion for  the  distance  of  the  centre  from  the  origin,  viz. 

x=-' — = • 

s  s 

But  when  the  line  is  not  symmetrical  with  respect  to  the  axis, 
then  we  must  determine  the  Y  of  the  centre  of  gravity  as  well  as 
the  X,  and  this,  by  the  second  of  equation  (A),  is 

s  8  dx 

is  given  in  terms  of  x  by  the  equation  of  the  line. 

,  Problem  I. — To  determine  the  centre  of  gravity  of  a  given 
straight  line. 

fxdx      7>  oc^ 

In  this  case  we  have  X=^- = =5  x,  therefore,  represent- 

x  X 

ingthe  whole  line  by  a,  we  have,  when  x=a,X=\  a,  so  that  the 
centre  of  gravity  is  at  the  middle  point,  as  indeed  is  obvious  without 
calculation. 

Problem  II. — To  determine  the  centre  of  gravity  of  the  contour 
of  any  polygon. 

Let  us  represent  the  sides  of  the  polygon  by  Pj,  Pj?  Pg*  <fec. 
and  let  the  co-ordinates  of  the  angular  points  be 

then  the  co-ordinates  of  the  middle  points  of  the  sides  will  be 
^\+x^  y^+y^ .  a^2+a^3  3/2+^3 .  ^^ 
2      '       2       '        2      '       2       ' 
and  these  points  are,  by  last  problem,  the  several  centres  of  gravity 
of  the  sides,  so  that  we  now  have  to  find  the  centre  of  gravity  of  the 
weights  P,  Pj,  Pj,  &c.  acting  at  these  points,  and  for  this  we  have 
the  equations 

2  /^p  _j_p    f  p  _|_ p  ■) 

Y_  Pt  {y,+y.)+^Xy.+y,)+'P.  Jy-^-i-yS-^  -  •  •  Pn  (y.+yi). 

2(P,  +  P,  +  P3+ p.) 


CENTRE    OF    GRAVITY.  63 

It  is  obvious  that  if  the  polygon  be  regular,  the  middle  point,  or 
the  centre  of  the  inscribed  circle,  will  be  the  centre  of  gravity,  and 
•f*!'  Pa'  P3'  *^^'  ^^^^^  ^^  ^^^  equal ;  hence  if  the  polygon  have  n  sides 

an  equation  which  expresses  this  geometrical  property,  viz.  that  if» 
from  the  corners,  and  from  the  centre  of  a  regular  polygon,  perpen- 
diculars to  any  line  in  its  plane  be  drawn,  the  sum  of  the  perpendicu- 
lars from  the  corners  will  be  equal  to  as  many  times  that  from  the 
centre  as  there  are  sides  to  the  polygon. 

Problem  III. — To  determine  the  centre  of  gravity  of  a  circular  arc 
BAC=2s  (fig.  35).  The  equation  of  the  curve  referred  to  the  axes 
AX,  AY  is 

y^=2rx — a;2 

'  '  <^^     ^2rx—x^  '  \      "^rfx^       \^2rx—x'^ 
•'•  ^=T/v7^r^=y ^-^'^^'^^^^'+'^  (Int.Calc.pm-S). 

s  ^         ^-^  J      s  s 

so  that  the  distance  of  the  centre  of  gravity  from  the  centre  of  the 

circle  is  a  fourth,  proportional  to  the  arc,  the  radius,  and  the  chord 

of  the  arc. 

When  the  arc  is  a  semicircle  the  chord  is  double  the  radius,  and 

then 

2r  r 

0G= = -=  -63662^ 

3  .  141593  . .      1  .  57079 

and  when  it  is  a  whole  circle,  then,  y  being  =  0,  OG  is  0,  as  we 
otherwise  know  to  be  the  case. 

It  may  be  here  observed,  that  in  this  solution  we  have  not  sought 
for  the  arbitrary  constant  necessary  to  complete  the  above  integral, 
nor  need  we  seek  for  it  in  any  case,  because  it  is  always  the  definite 
integral  that  we  want.  Since  the  body  whose  centre  of  gravity  we 
require  is  necessarily  limited,  it  is  between  its  limits  that  our  integral 
is  to  be  taken,  and  thus  limited  it  can  require  no  correction,  (Int. 
Calc.  p.  90-1.) 

Problem  IV. — To  determine  the  centre  of  gravity  of  the  arc  of  <i 
cycloid. 

The  diflferential  equation  of  this  curve  is,  (see  Int.  Calc.  p.  115,) 


64  ELEMENTS    OF    STATICS. 


dy        WT—y 


dx     \       y 
where  r  is  the  radius  of  the  generating  circle 

For  the  whole  curve  y^2  r,  so  that  the  distance  of  the  centre  of 
gravity  G  from  the  vertex  is  equal  to  one  third  of  the  diameter  of 
the  generating  circle. 

Determination  of  the  Centre  of  Gravity  of  a  Plane  Area. 

(43.)  When  the  body  is  considered  as  a  plane  area  w,  the  first  of 
the  equations  (A)  becomes,  by  substituting  u  for  B, 
fxdu      fxydx 

A= = — r— J .   .   .   .  (  1  U 

which  is  of  itself  sufficient  to  determine  the  centre  when  the  axis 
passes  tlirouffh  it,  that  is  when  this  axis  divides  the  figure  into 
symmetrical  halves.  But  suppose  we  require  the  centre  of  gravity 
of  the  part  ii  on  one  side  the  axis,  that  is  of  the  area  ABD  (fig.  36), 
then,  in  addition  to  X  =  AM,  we  must  also  know  Y  =  MG. 

Now  P  being  any  point  (x,  y),  we  are  not  to  substitute  in  the  ex- 
pression for  Y,  instead  of  dB,  the  value  ydx,  which  we  put  in  the 
expression  for  X  ;  because  now  we  want,  agreeably  to  the  general 
investigation,  not  the  differential  of  APN,  but  that  of  APN'D  ;  the 
former  was  correctly  expressed  by  PN  .  dx,  but  the  latter  must  be 
PN'  .  dy=(AJ) — AN)  dy=(a — x)  dy,  a  being  the  distance  of  the 
bounding  ordinate  from  the  origin  ;  hence 
^J-ydu^f{a—x)ydy 

«  fi.a—x)  dy  ^  ^' 

It  appears  then  that  the  expression  for  du  depends  upon  the  manner 
in  which  we  conceive  u  to  be  generated,  that  is  whether  we  consi- 
der the  increment  \ii  to  be  paralel  to  the  axis  of  y  or  to  the  axis  of 
'  X,  or,  which  is  the  same  thing,  whether  we  consider  in  the  equation 
of  u  the  independent  variable  to  be  x  or  y.  We  may,  however,  ex- 
press du  in  a  form  which  will  leave  it  optional  with  us  which  shall 
be  the  independent  variable  ;  this  form  is  du=fdx  dy,  where  the 
integral  sign  applies  to  either  of  the  difi'erentials  under  it,  regard 
being  had  in  the  integration  to  the  limits  between  which  the  va- 
riable is  comprised.  Thus,  in  the  case  we  are  considering,  if  we 
take  the  integral  sign  to  apply  to  dx,  and  integrate  between  the  limits 
r=a  and  x=x,  then  the  above  equation  is  the  same  as  du=(a — x) 
dy ;  but  if  we  consider  the  integral  sign  to  apply  to  dy,  then  the  inte- 


CENTRE    OF    GRAVITY.  65 

gral  y  being  between  the  limits  y=y  and  2/=0  (see  fig.  36),  the 
above  expression  is  the  same  as  du=ydx.  Hence,  by  writing  du 
in  the  above  general  form,  the  expressions  for  X  and  Y  will  be 

ffdydx'  ffdy'dx 

in  which  the  integrations  may  be  performed  in  any  order.     If  in  the 

first  of  these  we  integrate  for  3/,  first,  we  shall  have  the  form  (1) 

above  :  and  if  we  integrate  the  second  for  x  first,  we  shall  have  the 

1  fu^dx 
form  (2),  but  if  we  integrate  this  for  y  first,  then  Y=  "  -^     —  .  (3). 

Jydx 

When  the  area  is  bounded  by  straight  lines  only,  the  case  will  be 

too  simple  to  require  the  aid  of  the  calculus,  as  in  the  first  of  the 

following  problems. 

Problem  V. — To  determine  the  centre  of  gravity  of  a  plane  tri- 
angle ABC  (fig.  37). 

Bisect  any  side  as  AB  by  a  line  CD  from  the  opposite  angle,  then 
the  centre  of  gravity  must  be  in  this  line  ;  for  it  will  bisect  any  line 
a  h  whatever  drawn  parallel  to  AB,  so  that,  taking  any  point  in  da, 
there  shall  always  be  a  corresponding  point  in  (/  b,  equidistant  from 
CD,  or  from  a  plane  through  CD  perpendicular  to  the  plane  of  the 
triangle ;  hence  the  sum  of  the  moments  on  one  side  this  plane  will 
be  equal  to  the  sum  of  the  moments  on  the  other  side  ;  hence  the 
centre  of  gravity  being  necessarily  in  the  plane  of  the  figure,  must 
be  in  the  line  CD.  In  like  manner,  if  AC  be  bisected  by  the  line 
BE,  the  centre  will  lie  in  this  line,  hence  it  is  at  the  point  G  where 
it  intersects  the  former. 

We  might  immediately  infer  from  this,  that  the  three  lines  from 
the  angles  of  a  triangle  bisecting  the  opposite  sides  necessarily  meet 
in  a  point.  To  determine  the  distance  of  this  point  from  C  let  us 
draw  ED,  which,  as  it  bisects  the  two  sides  AB,  AC,  must  be  paral- 
lel to  CB  and  equal  to  half  of  it ;  hence  the  triangles  EGD,  BCD  are 

BC       CG 

similar  .•.——-=-— --=2  .-.  CG=|C|!I;  to  express  this  distance 
ED       GU)  ^  ^ 

analytically,  put  a,  b,  c,  for  the  sides  respectively  opposite  to  the 

angles  A,  B,  C,  and  e  for  the  line  CE,  then,  (Geometry,  p.  38-9,j 

2e=^\2a''-\-2h^—c^\  .-.  CG==i  ^{2«2-f  26^—0^} 

in  like  manner  BG=^  ^Z \'2a'' +2c''—b''\ 

kG=\  y/l^b'^+^c^—cv'] 

Adding  together  the  squares  of  these  expressions  there  results  the 

equation  3  (AG^+BC-f  CG«)  =AB2  +  BC2+CA'^ ;  that  is,  in  any 

plane  triangle  the  sum  of  the  squares  of  the  sides  is  equal  to  three 

times  the  sum  of  the  squares  of  the   distances  of  the  vertices  from 

tlie  centre  of  gravity  of  the  triangle. 

f2  9 


66  ELEMENTS  OF  STATICS. 

If  three  equal  bodies  be  placed  at  the  vertices  of  a  triangle,  their 
centre  of  gravity  will  coincide  with  tlie  centre  of  gravity  of  the  tri- 
angle ;  for  the  centre  of  gravity  of  the  two  equal  bodies  A,  B  wiU 
be  at  D,  and  the  weight  at  this  point  will  be  2  A  ;  hence,  joining 
D,  C,  the  centre  of  gravity  of  2  A  and  C  =  A  will  be  a  point  G, 

^    ^     2  A       CG 
such  that  — -—  ^  T^-TT  . 

A  (jrD 

Again,  the  centre  of  gravity  will  still  be  the  same  point  if  the 
equal  weights  be  placed  at  the  middle  points  D,  E,  F  ;  for  the  mid- 
dle M  of  DE  will  l)e  the  centre  of  gravity  of  two  of  them,  and  the 
centre  of  these  and  the  third  will  be  a  point  G,  such  that  MG  = 
5GF  and  MG  is  equal  to  hGF,  because,  by  similar  triangles, 

BF       GF 
EGM,BGF,  j^=^,  =  2. 

The  centre  of  gravity  of  any  plane  polygon  may  be  found  by  first 
finding  the  centres  of  gravity  of  its  component  triangles,  and  then 
the  common  centre  of  gravity  of  all  these  points  loaded  with  their 
respective  triangles. 

Problem  VI. — To  determine  the  centre  of  gravity  of  a  circular 
segment  (fig.  38). 

From  the  equation  of  the  curve,  taking  the  centre  for  origin,  and 
putting  OD  =  a,  we  have  y=^\r^  —  x^  \  3 

'  '  u  u  u  ' 

If  the  segment  is  a  semicircle  a  =  0,  and  u  =  1-57079  r",  hence,  in 

this  case,  X= J^  =  ^-^  =  -42441  r. 

Problem  VII. — To  determine  the  centre  ofgravity  of  the  common 
parabola  (fig.  39). 

From  the  equation  of  the  curve,  y  =  \/[pa:| 

_f  xy  dx  _  /xf  dx       3  3 

fydx  fxk  dx      b  5 

If  we  require  the  centre  of  gravity  of  the  semi-parabola  ABa:,  then 
Y,  which  for  the  whole  parabola  is  0,  will  have  to  be  calculated  by 
the  second  formula  (3),  that  is,  we  shall  have 

^^     fy^dx        i    fxdx  3      .c      ,       3  3  „ 

^=2ijdi=P'    27^rrf^=F^''^^^=T2^  =  ¥^^- 

Determination  of  the  Centre  of  Gravity  of  a  Surface  of  Revolution. 

(44.)  Let  the  axis  of  x  be  the  axis  of  revolution,  then  it  will  pass 
through  the  centre  of  gravity  of  the  body,  and  therefore  the  first  of 


CENTRE    OF    GRAVITY.  67 

the  general  equations  (A)  will  be  sufficient  to  determine  it.     Putting 
S  for  the  surface,  this  equation  becomes,  (see  Int.  Cede.  p.  138,) 


f  xd^        fxy  (Is 

J\.  — ^  -  * 


/^W'^^'^-'^ 


S  fyds 


fy 


dx 


Problem  VIII. — To  determine  the  centre  of  gravity  of  a  spheric 
surface  (fig.  40). 

By  the  circle,  x^-\-y'''=r'^ 

dy^      x'^         ,  ,  fxyds       f  xdx 

•'•  -T-7  =  —^  •'•  y(^^  =  ^d^  •'•  -"h^i =r,—  • 

dx^      y^      ^  fyds         f  dx 

Hence,  integrating  between  x=r  and  a:;=OC=a,  we  have 

X=il!:!irj^=i(r+  a)  =0G ; 
r  —  a 

hence  G  is  at  the  middle  of  CB. 

Problem  IX. — To  determine  the  centre  of  gravity  of  a  conical 
surface  (fig.  41). 

Taking  the  centre  of  the  cone  as  the  origin,  we  have  for  the  re- 
volving line  OB  the  equation 

y=ax.'.-^=a^.-.yds=a  .^{l+a^]  dx 

fxyds^fx^dx^2^^  1.0C. 
fy  ds       fx  dx      3  3        * 

Hence  G  is  at  the  distance  of  one  third  the  height  from  the  base. 

Determination  of  the  Centre  of  Gravity  of  a  Solid  of  Revolution. 

(45.)  Putting  V  for  B  in  the  general  expression  for  X,  and  we 

fx  dY       fxy^  dx 
have  lor  any  volume  oi  revolution  X  =  =^-^r;: —  =  -  ,  „  ,     • 
•^  V  fy'^  dx 

Problem  X. — To  determine  the  centre  of  gravity  of  a  segment  of 
a  spheroid  (fig.  42). 

Taking  the  origin  at  A,  the  vertex  of  the  generating  ellipse,  we 
have  for  its  equation  ■  .:   .   f  ■ 

„_6^  fxy^  dx     f(2ax — x^)  xdx 

y-=-,{^ax—x  )  .-.  X=-p-^-=   j-^2ax'-x')  dx  ' 

.^      ^ax^ — kx*     8ax—3x^ 

that  is  AG=^— ^ — r—  =  — — • 

ax^ — la^        12  a — 4x 

For  a  hemispheroid  we  have  a:=aand  X=fa. 


08  ELEMENTS   OF    STATICS. 

As  the  foregoing  expression  for  AG  is  independent  of  b,  it  re- 
mains the  same  when  b=a,  or  >vhen  the  body  is  a  sphere,  so  that 
if  a  sphere  be  described  on  either  axis  of  a  spheroid,  any  segments 
cut  off  by  a  plane  perpendicular  to  this  axis  will  have  the  same  cen- 
tre of  gravity. 

(46.)  The  general  formula  employed  in  this  article  will,  after  a 
slight  modification,  serve  to  determine  the  centre  of  gravity  of  any 
volume  generated  by  the  motion  of  a  varying  surface  along  a  fixed 
axis,  perpendicular  to  its  plane,  and  passing  through  its  centre  of 
gravity,  as  the  axis  of  x,  provided  only  that  in  every  position  this 
surface  is  the  same  function  of  x.  For,  calling  the  generating  plane 
K,  the  differential  ^/B  of  the  body  generated  will  be  K  dx,  (Int. 
Calc.  p.  144,)  and  hence  the  formula  in  last  article  will  be 

/K  xdx 

JKdx 
We  shall  give  two  examples  of  the  application  of  this  form  of  the 
general  equation. 

Problem  XI. — To  determine  the  centre  of  gravity  of  a  pyramid 
or  cone. 

Let  AB  be  the  axis,  which  call  b,  and  the  area  of  the  base  HA, 
a  ;  then,  since  the  area  of  any  two  sections  perpendicular  to  the  axis 
are  as  the  squares  of  their  distances  from  A,  we  have 
b^_a_      Tz_<^^'' 

substituting  this  value  of  K  in  the  formula  above,  there  results 
fKx  dx  _  fx^  dx  _  3a,-*  _   3 
~  /K  dx    ~  /r»  dx   ~~4x^~  T^' 
hence  AG  is  equal  to  I  the  altitude  of  the  cone  or  pyramid. 

If  the  solid  is  the  frustum  or  trunk  of  a  cone  or  pyramid  of  which 
the  distances  of  the  two  ends  from  A  are  respectively  b',  b,  then,  in 
the  expression  for  x,  we  must  integrate  between  these  values  of  x, 
that  is 

fl.Kxdx     f\.x^dx      3    b*—b'* 


/*,  K  rfa;     /*,  x""  dx     4    b^—b'^ 

Problem  XII. — To  determine  the  centre  of  gravity  of  any  seg- 
ment of  an  ellipsoid. 

Calling  the  principal  semiaxes  a,  b,  c,  the  equation  of  the  surface 
is  a''6^2;'+aV^y''^-6^c'a:*=a''6"c^  and  the  equation  of  a  section 
at  the  distance  x  from,  and  parallel  to,  the  plane  of  yz,  is 

b'  z'+c'  y'= ^-- ^ 


CENTRE  OF  GRAVITY.  69 

irom  which  we  get  these  functions  of  x  for  the  semiaxes  of  the  ge- 
nerating ellipse,  viz. 

,,     bs/laJ'—xH      ,     c^\d'  —  x''\ 

0  = ■ -1  C  = ■ -1 

a  a 

and  therefore  the  area  of  the  generating  surface  in  any  position  is, 

(Int.  Calc.  p.  124,)  rtb' c' =^  {d'—x^)=K 


X- 
fKxdx_fx  (a^— a?==)  dx_h  a^  x''--ix*_  Ga^— Sa?^ 


X. 


fKdx  f(a^—x^)dx  a^  x—^  x^  12a''~4x^ 
This  expression  being  independent  of  6  and  c,  shows  us  that  ellip- 
soids having  one  common  axis,  have  a  common  centre  of  gravity  for 
all  the  segments  cut  off  by  a  plane  perpendicular  to  that  axis  ;  which 
is  an  extension  of  the  property  noticed  at  (45). 

(47.)  Having  now  given,  in  succession,  all  the  most  usual  for- 
mulas for  the  determination  of  the  centre  of  gravity,  together  with 
their  practical  illustration,  we  shall  terminate  by  merely  writing 
down  the  forms  which  the  general  equations  (A)  take  when  we  wish 
to  apply  them  to  a  surface,  or  a  volume  which  is  not  of  revolution, 
nor  yet  symmetrical  with  respect  to  an  axis. 

The  general  expression  for  the  differential  of  a  surface  S,  referred 
to  three  rectangidar  planes,  is,  (Int.  Calc.  p.  151,) 

where  it  is  optional  for  us  to  consider  the  differential  dS  to  be  taken 
either  with  respect  to  x  or  with  respect  to  y ;  hence  putting  this  for 
dB  in  the  general  equations  referred  to,  or  rather  writing  the  coeffi- 
cients under  the  radical  in  the  more  abridged  form  p',  q',  we  have 
these  expressions  for  the  co-ordinates  of  the  centre  of  gravity  of  the 
surface, 


ff  X  y/  l+p'^+q"".  dx  dy 
ffs/  T+p^+q'\  dx  dy 


X^j./j^y_ 


Y^//y  ^  l+p'^+q'^-  dx  dy 
ff^  l+p'^'-^-q'^.  dx  dy 

ffx/l+p"'+q'^'  dx  dy' 
Again,  the  general  expression  for  the  differential  of  a  volume  V  is, 
(Int.  Calc.  p.  148,)  dY=f  z  dy  dx;  in  which  it  is  optional  whe- 
ther we  consider  the  differentiation  to  be  performed  with  respect  to 
the  independent  variable  x  or  y;  but  we  may  render  this  expres- 
sion still  more  general,  as  well  as  more  symmetrical  by  putting  fdz 
for  z,  for  then  the  differential  dY=ff  dz  dy  dx  may  be  considered 


70  ELEMENTS  OF  STATICS. 

as  taken  relatively  to  either  of  the  three  variables  .t,j/,  r,  we  please. 
Hence  the  expressions  for  the  co-ordinates  of  the  centre  of  gravity 
of  the  volume  are 

fffdz  dy  dx         ff  ^  dx  dy 

y^Mj/JliIiu!^  or//-y-  '^y  ^^ 

fff  dz  dy  dx        ff  z  dy  dx 

2.=IILll^-^y—  or  iff^^^y^^^ 

fffdz  dy  dx         ffzdydx  ' 
On  Guldin's  Theorem,  or  the  Centrobaryc  Method. 

(48.)  The  expressions  for  Y,  at  articles  (42)  and  (43),  furnish  a 
very  remarkable  theorem  for  the  determination  of  the  surfaces  and 
volumes  of  bodies  of  revolution.  The  expressions  referred  to  im- 
mediately give  the  equations 

2  rt  Ys=2  rt  fyds  and  2  rt  Y  f  ydx=7t  f  y^  dx ; 
the  former  relating  to  the  curve  line,  the  latter  to  the  surface.  Now 
2rtY  is  the  circumference  of  which  Y  is  the  radius ;  it,  therefore, 
expresses  the  circumference  which  would  be  described  by  the  cen- 
tre of  gravity  of  the  line  5  if  it  were  to  revolve  round  the  axis  of  x : 
but  2  ft  fyds  expresses  the  area  of  the  surface  which  would  be 
engendered  by  this  revolution:  hence,  1.  The  surface  getierated  by 
the  revolution  of  a  curve  round  an  axis  is  equal  to  the  length  of 
that  curve  multiplied  by  the  circumference  described  by  the  centre 
of  gravity. 

Again,  2  rt  Y,  in  the  second  of  the  above  equations,  being  equal 
to  the  circumference  which  would  be  described  by  the  centre  of  gra- 
vity of  the  surface  s,  if  it  were  to  revolve  round  the  axis  of  x,  and 
nfy^dx  being  the  expression  for  the  very  volume  which  would 
thus  be  generated,  it  follows  that,  2.  Tlie  volume  generated  by  the 
revolution  of  a  plane  surface  round  an  axis  is  equal  to  the  area  of 
that  surface  multiplied  by  the  circumference  described  by  its  centre 
of  gravity. 

These  two  propositions  comprise  the  theorem  of  Guldin,  and 
their  application  to  the  determination  of  the  surfaces  and  volumes 
of  bodies  constitutes  the  Centrobaryc  method.  By  this  method  we 
see  that  when  we  know  the  length  of  the  generating  line,  or  the  area 
of  the  generating  surface,  as  also  the  distance  of  its  centre  of  gravity 
from  the  axis  of  revolution,  the  value  of  the  surface,  or  solid  gene- 
rated either  by  a  whole  or  a  partial  revolution,  may  be  at  once  found. 
Also  any  two  of  these  three  things  being  given,  viz.  the  generatrix, 
the  distance  of  its  centre  of  grarity  from  the  axis,  and  the  magnitude 
generated  being  given,  the  third  may  be  found. 


CENTRE    OF    GRAVITY.  71 

As  an  example  of  this  method,  suppose  we  wanted  to  know  the 
volume  of  a  paraboloid  of  revolution  :  Let  a  be  the  axis  or  height  of 
the  generating  semi-parabola,  and  b  its  base  ;  then  the  distance  of 
its  centre  of  gravity  from  the  axis  §6,  so  that  the  circumference  ge- 
nerated by  this  centre  is  Ibn.  Again,  the  area  of  the  generating  sur- 
face is,  (Int.  Calc.  art.  67,)  §c6,  consequently,  multiplying  these  two 
quantities  together,  have,  for  the  volume  sought, 

3  2  1 

V=— -  6  rt  x-^  ai  = -roi^  rt ; 

4  3  2 

this  volume  is,  therefore,  I  that  of  its  circumscribing  cylinder. 

Suppose  now  that  we  wished  to  determine  by  this  method  the  cen- 
tre of  gravity  of  a  semicircle  of  radius  r.  We  know  that  the  area 
of  this  semicircle  is  k  r^ri,  and  that  the  volume  of  the  sphere,  gene- 
rated by  it  is  ^  r^rt  ;  hence,  as  this  expression  must  be  equal  to  the 
former  multiplied  by  the  circumference  described  by  the  centre  of 
gravity  sought,  we  have  for  this  circumference,  the  value 

ill^=lr;  .•.|-r--2rt  =  -42441r=X; 
ir^  7t         3  3 

and  this  is  a  more  simple  way  of  determining  the  centre  than  that 

employed  in  problem  VI. 

We  shall  conclude  the  present  chapter  Avith  a  few  miscellaneous 

examples  for  the  exercise  of  the  student. 

1.  For  the  centre  of  gravity  of  a  parabola  of  the  nih.  order,  whose 

.      .          ,                 ,                  .       •    ^     2n+l 
equation  is  a"— i2/=x"  ;  the  expression  is  X=- -x. 

2.  For  the  distance  of  the  centre  of  gravity  of  a  semi-ellipse  whose 
axes  are  2a,  2b,  from  the  base  or  minor  axis,  the  expression  is 

3  rt  '  ..         -•'" 

■  '  2 

3.  For  a  paraboloid  of  revolution,  whose  altitude  is  a,  X=—a. 

o 

4.  For  a  segment  of  hyperboloid,  whose  altitude  is  a, 

_,      Sa4-Sx 

^  =  T^ 7-^' 

12a+4a; 

5.  The  convex  surface  of  a  conic  frustum,  or  trunk,  is  found  by 
Guldin's  theorem  to  be  equal  to  half  the  sum  of  the  circumferences 
of  the  ends  multiplied  by  the  slant  height.  "JX^  *  - 

SCHOLIUM. 

It  has  been  shown  in  the  outset  of  this  chapter,  that  for  a  body  to 
be  supported,  it  is  absolutely  necessary  that  its  centre  of  gravity  lie 
in  a  vertical  line,  passing  through  the  base  on  which  the  body  stands ; 


72  ELEMENTS    OF    STATICS. 

or  if  the  boily  stand  on  props  or  legs,  this  vertical  line  must  pass 
through  the  area,  whirh  a  string  stretched  round  these  legs  would 
enclose.  The  space  thus  enclosed  by  the  feet  of  the  human  body 
is.  obviously,  but  small,  and  M'hen  we  consider  the  very  various 
positions  in  standing,  stooping,  walking,  &;c.  which  we  can  easily 
and  safely  assume,  and  the  great  rapidity  with  which  we  can  pass 
from  any  one  of  these  positions  into  another,  we  cannot  fail  to  be 
impressed  with  the  wisdom  and  bounty  of  the  great  Creator,  who, 
by  such  admirable  disposition  of  the  limbs  and  joints  of  the  body 
lias  rendered  this  small  space  sufficient  for  its  support  in  all  these 
various  attitudes.  A  person  in  danger  of  falling  experiences  an 
irresistible  impulse  to  overtake,  as  it  were,  tlie  point  where  the  line 
of  direction  seeks  to  meet  the  horizontal  plane,  and  hence  the  long 
strides  which  he  is  impelled  to  make  when  violently  pushed  in 
the  back,  the  feet  endeavouring  to  overstep  the  point  alluded  to ;  so 
Avhcn  standing  and  inclining  the  body  forward,  till  the  line  of  di- 
rection is  about  to  fall  beyond  our  toes,  we  cannot  help  putting  for- 
M'ard  our  foot  to  overstep  it.  Children  who  have  not  the  same 
command  over  their  limbs  as  grown  persons,  and  who  have,  more- 
over, a  less  sense  of  danger,  do  not  always  use  the  best  means  to 
recover  their  stability  when  about  to  fall,  but  then  in  the  more 
hazardous  circumstances  they  usually  take  care  to  secure  for  them- 
selves a  more  spacious  base  than  grown  persons ;  thus  in  walking 
up  or  down  stairs,  we  commonly  see  children  employ  both  their 
hands  and  feet,  thus  securing  for  themselves  a  very  considerable 
base,  out  of  which  the  line  of  direction  is  not  easily  forced. 

If  a  body  rest  on  a  single  point,  it  is  necessary  that  the  vertical 
line  through  it  should  pass  also  through  the  centre  of  gravity,  but 
"  in  certain  cases  a  body  resting  upon  a  single  point  may  yet  have  a 
disposition  to  recover  from  any  derangement,  and  to  resume  its  ver- 
tical position.  Thus  if  the  base  be  a  plane,  and  the  bottom  of  the 
body  rounded,  but  such  that  the  centre  of  gravity  lies  below  the  centre 
of  curvature,  the  mass  may  rock  backwards  and  forwards,  but  will 
soon  regain  its  erect  site.  Let  O  (tig.  43,)  be  the  centre  of  the  incur- 
vation at  the  end  of  the  body,  and  G  or  o-  its  centre  of  gravity  lying 
in  the  axis  AO.  Conceive  the  body  to  be  rolled  on  its  horizontal 
plane  from  A  to  A',  the  point  which  touched  A  will  merge  into  o, 
and  the  axis  will  come  into  the  position  aO'.  Now  if  the  centre  of 
gravity  G  stood  above  0,  it  would  evidently  in  the  position  G'  lean 
beyond  the  vertical  A'O',  and  the  body  would  fall  over  :  but  if  the 
centre  of  gravity  were  at  g  below  0,  it  would  still  in  changing  to 
g'  lie  within  the  vertical  A'O',  and,  consequently,  the  body  would 
roll  back  to  its  first  position."     Leslie's  Natural  Philosophy,  p.  55. 

It  may  be  remarked,  that  when,  as  in  the  case  just  adduced,  the 
body  resists  the  tendency  to  overturn  it,  and*  returns  to  its  first  po- 


EQUILIBRIUM    OF    A    SOLID    BODY.  73 

sition  of  equilibrium,  this  equilibrium  is  called  stable  ^  but  when  it 
yields  to  every  tendency  to  overturn  it,  and  falls,  the  equilibrium 
from  which  it  has  been  disturbed  is  called  unstable.  The  equili- 
brium of  an  ellipse  resting  on  the  extremity  of  its  minor  diameter 
is  stable,  but  the  equilibrium  when  it  rests  on  the  extremity  of  the 
major  diameter  is  unstable. 


CHAPTER  III. 

ON    THE    EQUILIBRIUM    OF    A    SOLID    BODY    ACTED    UPON    BY    FORCES 
APPLIED    TO    DIFFERENT  POINTS  AND  IN  DIFFERENT  DIRECTIONS. 

(49.)  We  come  now  to  consider  a  body  or  system  of  material 
points  connected  together  in  an  invariable  manner,  when  in  a  state 
of  equilibrium  from  the  action  of  any  system  of  forces  whatever, 
and  to  determine  what  the  conditions  are  which  must  necessarily 
characterize  such  a  system.  The  reasonings,  therefore,  of  this 
chapter  will  be  so  general  that  the  results  to  which  they  lead  will 
comprehend  in  them  all  the  particulars  hitherto  deduced  respecting 
the  equilibrium  of  forces  acting  under  certain  restrictions,  wdiether 
through  the  intervention  of  a  solid  body,  or  upon  a  single  point. 
The  more  important  of  these  particular  deductions  are,  however, 
essential  to  the  establishment  of  the  general  theory,  so  that  we  are 
not  to  expect  that  the  discussion  of  the  general  proposition,  on 
which  we  are  about  to  enter,  will  be  independent  of  its  particular 
cases  already  considered,  but  that  it  will  on  the  contrary  be  in  a 
great  measure  founded  upon  them.  We  shall  find  it  convenient  to 
divide  this  proposition  into  two  parts,  taking  first  the  case  where 
the  component  forces  all  act  in  one  plane,  and  afterwards  considering 
them  to  act  without  restriction. 

I.  When  the  forces  are  all  situated  in  one  plane. 

(50.)  Let  the  plane  of  the  forces  P,  P^,  P^,  &c.  be  taken  for  the 
plane  of  xy,  and  let  their  points  of  application  be  (x,  y),  (x^,  y^), 
(Xg,  2/2),.  &c.,  also  let  the  inclinations  of  the  forces  to  the  axis  of  x 
be  as  usual,  a,  a^,  02,  &c,,  and  to  the  axis  of  y,  j3,  j3-^,  8^,  &;c. ; 
these  latter  angles  being  the  complements  of  the  former.  Let  now 
each  force  be  decomposed  into  two  parallel  to  the  axis,  and  we  shall 
thus  have  instead  of 

P  the  components  P  cos.  a,    P  cos.  /3 
Pj  .         .         P  COS.  ttj,  P  cos.  /Sj 

Pa     .         •         .     P  COS.  ttj,  P  COS.  ^2 
&-C.       •  &c. 

G  10 


(0; 


74  ELEMENTS    OF    STATICS. 

SO  that  the  points  are  now  acted  upon  by  two  systems  of  parallel 
forces,  and,  supposing  that  each  system  has  a  single  resultant,  it 
will  be  parallel  to  the  components,  and  it  follows  that  the  original 
system  may  be  replaced  by  two  forces  X  and  Y,  of  which  the  in- 
tensities are 

X=P  COS.  a-fPi  COS.  ttj-fPj,  cos.  03-f&.C.  > 
Y  =  P  cos. /3-j-Pi  cos.  /3i-|-P2  COS.  iSj+'Stc.  5  ' 
thus  when  we  know  the  right-hand  members  of  these  equations,  as 
we  are  here  supposed  to  do,  we  know  also  the  intensities  and  direc- 
tions of  the  two  forces  X,  Y,  but  not  as  yet  their  points  of  applica- 
tion. 

To  determine  these  let  y'  be  the  distance  of  the  force  X,  from  the 
axis  of  X,  to  which  it  is  parallel,  and  x'  be  the  distance  of  Y  from 
the  axis  of  y,  then  we  know  (38)  that 

Xy'=yVcos.a+y^V^  cos.a^+y^  P^  cos.a^-f  &c.  ?  ,  . 

Yx'=ar  Pcos. /3-f-a;i  P,  cos./3i-far2  Pacos.^g-f&c.  5  '  •  '  •\'^J'' 
from  which,  as  the  values  of  x,  a?,....,  y,  y,....  are  supposed  to  be 
known,  x'  and  y'  become  known,  and  two  lines  drawn  parallel  to 
and  at  these  distances  from  the  axes  will  coincide  with  the  direc- 
tions of  the  forces  X,  Y,  and  as  a  force  may  be  applied  at  any 
point  of  its  direction,  we  may  consider  the  intersection  of  these  two 
lines  to  be  the  common  point  of  application  of  the  two  forces  X,  Y, 
so  that  the  original  system  is  thus  reduced  to  two  determinate  forces, 
acting  on  a  determinate  point  in  known  directions  ;  and,  lastly,  these 
are  reducible  to  a  single  determinate  force  R,  by  the  equations 

R=^/fX^-f  YM  cos.  a=|,  cos.  b=l  ....  (3); 

a  and  b  being  the  inclinations  of  R  to  the  axes  of  x  and  y. 

It  is  thus  proved,  that  when  a  system  of  forces,  acting  with  given 
intensities,  and  in  given  directions,  upon  a  given  system  of  points 
invariably  connected,  has  a  single  resultant,  its  intensity,  direction, 
and  point  of  application  may  all  be  determined  by  means  of  the 
given  quantities. 

(51.)  It  should  be  remarked  here,  that  the  equations  (2)  which 
enable  us  to  determine  the  point  (x',  y')  of  application  of  the  result 
ant,  after  we  have  found  X  and  Y,  furnish  us  with  more  information 
than  we  absolutely  require,  for  it  would  be  quite  sufficient  for  us  to 
know  the  situation  of  any  point  through  which  the  resultant  passes, 
for  knowing  this  the  position  and  intensity  of  it  would  be  given  by 
equations  (3),  and  the  effect  of  it  will,  we  know,  be  the  same  to 
whatever  point  it  be  applied.  Again  the  position  thus  determined 
ought,  obviously,  to  be  independent  of  the  co-ordinates  x,  y  ;  a?,,  y^ ; 
(fee.  because  it  would  remain  the  same  to  whatever  points  the  forces 


EQUILIBRIUM    OF    A    SOLID    BODY.  75 

P,  P  ,  <fcc.  are  applied,  provided  we  take  them  in  their  directions. 
What  is  here  said  amounts  to  this,  viz.  that  we  ought  to  be  able  to 
determine  thy  perpendicular  distance  of  the  resultant  from  a  given 
fixed  point,  when  we  know  the  perpendicular  distances  of  the  com- 
ponents from  that  point,  and  this  determination  we  shall  be  able  to 
effect  by  means  of  the  equations  (2). 

The  given  fixed  point  from  which  the  distances  of  the  directions 
of  the  several  forces  are  to  be  estimated,  we  shall,  for  simplicity, 
consider  to  be  the  origin  and  the  product  of  each  force  by  the  per- 
pendicular on  its  direction,  we  shall  call  the  moment  of  that  force 
with  respect  to  the  proposed  point.  This  being  premised,  let  us 
substitute  for  X  and  Y,  in  equations  (2),  their  values  (3),  viz. 
Rcos.aand  R  cos.  b,  and  then  subtract  the  one  equation  from  the 
other,  and  we  shall  thus  have 

R  (?/'  cos.  O' —  x'  cos.  6)=P  {y  cos.  a  —  x  cos.  j3)  + 

Pi  (2/1  ^0^-  «■!  —  ^1  ^^^-  ^1)+  ^^^ 
Let  us  inquire  into  the  meaning  of  the  first  side  of  this  equation. 
Suppose  m  (fig.  44)  to  be  the  point  of  application  of  the  resultant 
'R=:m  n,  then      mB=a:',   mA  =  y',    and   let   perpendiculars    be 
drawn  from  0,  A,  B,  on  the  resultant,  then  it  is  plain  that  ?/'cos.a 
=A.p,  x'  COS.  ^=B  q  ;  also,  if  Os  be  drawn  parallel  to  7np  the 
triangles,  m^/B,  OsA,  will  be  in  all  respects  equal,  and,  therefore, 
As=By,  therefore  Aja  —  65'= Or,  which  call  r,  consequently 
R  (?/'  cos.  a  —  x'  cos.  b)='Rr^moment  of  R  .  .  .  .  (4). 
In  like  manner,  for  any  other  force  P„,  we  must  have 

Pn  (3/«  <^os.  a„  — x„  cos.  /3„)=P„;?„=momen?  o/P„ (5)  ; 

jo„  representing  the  perpendicular  from  O  on  the  direction  of  P„ ; 
hence  the  foregoing  equation  is  the  same  as 

nr=Fp+V,p^+F,p,+F,p,+  &c (6); 

so  that  the  moment  of  the  resultant  is  equal  to  the  sum  of  the  mo- 
ments of  the  components. 

In  the  figure  to  which  our  reasoning  has  been  applied  we  have  so 
taken  the  direction  of  the  force  R  that  the  angles  a  and  b  may  have 
positive  cosines :  but  the  inference  (4)  will  always  be  the  same 
whatever  be  the  directions  of  R,  thus  in  fig.  45,  where  cos.  b  is 
negative,  the  same  reasoning  as  that  employed  above  shows  us  that 
Or=Bq-\-Ap,  and  thus  the  same  conclusion  (4)  is  obtained. 

(52.)  The  foregoing  general  theorem  is  analagous,  in  terms,  to 
that  given  at  (38),  for  the  moments  of  a  system  of  parallel  forces 
with  respect  to  a  plane ;  but  the  student  must  carefully  observe  that 
there  is  no  analogy  between  the  theorems  themselves ;  for  the  mo- 
ment of  a  force,  with  respect  to  a  point,  is  altogether  distinct  from 
its  moment  with  respect  to  a  plane ;  this  latter  moment  depends  on 
the  point  of  application  of  the  force,  and  is  independent  of  its  direc- 


76  ELEMENTS    OF    STATICS. 

lion;  whereas  the  moment,  with  respect  to  a  point,  on  the  contrary, 

depends  on  the  direction  of  the  force,  and  is  independent  of  its 

point  of  application. 

The  second  side  of  the  equation  (6)  is  supposed  to  be  entirely 

known,  and  this  is  only  supposing  that  the  intensities  P,  Pj,  &c. 

are  known,  and  the  several  directions  in  which  they  act,  or  which 

is  the  same  thing,  the  equations  of  these  directions.     Thus,  suppose 

the  equation  of  P„'s  direction  is  known  to  be 

t"os.  /3„     ,  ,        , 

y= -x-\-b  .'.  b  cos.  a,=v  cos.  a„  —  X  cos.  3.., 

^     cos.  o„  "     -^ 

X  and  y  being  co-ordinates  of  any  point  whatever  in  the  given  di- 
rection ;•  if  we  take  any  particular  point  (x„,  y^)  we  have 

b  cos.  a„  =  i/„  COS.  o„  —  x„  cos.  ^3,, ; 
therefore,  by  equation  (5),  last  article  Pn=b  cos.  o„  ;  and  b  and 
COS.  a„  are,  by  hypothesis,  known  both  as  to  sign  and  quantity  ; 
hence  p^  is  known  both  as  to  sign  antl  quantity.  Hence  the  seconil 
member  of  the  equation  (6)  is  entirely  known  when  the  intensities 
and  directions  of  the  forces  are  known ;  also  R  is  known,  from  tlie 
same  data,  by  the  equations  (1)  and  the  first  of  (3)  in  article  (50), 
consequently,  r,  and,  therefore,  the  position  of  the  resultant  R  is 
known.  By  using  the  notation  employed  at  (39)  the  expression 
for  r  will  be 

and  the  equations  necessary  for  the  determination  of  the  resultant, 
both  in  intensity  and  direction,  will  be  those  marked  (1)  and  (6)  in 
last  article ;  that  is  the  three  following 

X  =  2  (P  COS.  o),  Y  =  2  (P  cos.  j3),  Rr=X  (P/)); 
the  first  two  giving  R=v/|X--f-Y*}. 

(53.)  When  the  system  is  in  equilibrium  then  we  have  X=0, 
Y=0,  and,  consequently,  R=0,  so  that  the  equations  of  equili- 
brium are  2  (P  cos.  a)=0,  S  (P  cos.  /3)=0,  2  (Pp)=0;  or,  in 
their  more  expanded  form, 

P  cos,  a  +  Pi  cos.  a^  +  Pg  cos.  aj  +  &C.  =  0") 

P  COS.  ^+Pj  cos. /3j+P3  cos.  ,3a+&;c.=0  I  .  .  (I). 

^P  +Pi/'i  +P2/'.  -f&c.=0j 
We  have  already  shown  how  the  signs  and  values  of  the  several 
products  in  the  last  of  these  equations  are  to  be  determined  from 
the  data  of  the  problem ;  but,  in  the  case  of  equilibrium,  it  some- 
times is  convenient  to  determine  the  signs  from  other  considerations 
less  analytical.  Let  us  suppose  the  origin  of  the  axes  A  (fig.  46), 
which  we  have  taken  for  the  centre  of  moments,  to  be  connected 
with  the  system  by  means  of  the  rigid  perpendiculars/),  p^,  p^,  &c. 
on  the  directions  of  the  several  forces  ;  then  the  whole  system  be- 


A    •!£- 


E<1UILIBRIUM  OF  A  SOLID  BODY.  77 

ino-  at  rest  the  point  A  will  be  at  rest,  and  may,  therefore,  be  con- 
sidered as  fixed.  Now  the  tendency  of  the  forces  P,  P^,  P^,  as 
marked  in  the  figure,  is  to  make  the  extremities  of  p,  p^,  p^,  on 
which  they  act,  to  turn  round  A  in  the  same  direction,  and,  conse- 
quently, the  tendency  of  the  whole  system  with  which  these  points 
are  invariably  connected  is  to  turn  round  A ;  to  prevent  motion, 
therefore,  the  remaining  forces  Pg,  P^,  &c.  must  impress  on  tlie 
system  an  equal  tendency  to  turn  round  A  in  the  opposite  direction  ; 
we  ought,  therefore,  to  attach  to  these  opposing  tendencies  contrary 
signs,  that  is,  if  we  consider  the  forces  P,  P^,  P^,  &c.  conspiring 
in  one  direction  to  be  positive,  we  ought  to  consider  the  forces  Py, 
P^,  &c.  conspiring  in  the  opposite  direction  to  be  negative.  In  the 
former  method  of  determining  the  sign  of  any  moment  P„  p^^,  we 
considered  P„  to  have  always  the  same  sign,  or  to  be  positive,  so 
that  the  moment  always  took  the  sign  of  p^.  If,  on  the  contrary, 
we  attach  the  sign  to  P„,  it  ought,  obviously,  to  be  in  conformity 
to  the  principle  just  stated,  and  then  we  must  consider  p^  to  be  al- 
ways positive 

(54.)  Of  the  preceding  equations  of  equilibrium  we  may  now 
remark  that  the  two  first  establish  the  condition  that  there  must 
exist  some  fixed  point.  A,  in  connexion  with  the  system,  round 
which,  however,  the  system  may  revolve  ;  the  thii-d  equation,  in 
conjunction  with  these,  must  then  establish  the  remaining  condition 
of  equilibrium,  viz.  that  there  can  be  no  motion  of  rotation.  If  the 
body  or  system,  on  which  the  forces  act,  be  not  entirely  free,  but 
be  only  at  liberty  to  move  in  any  direction  about  a  fixed  point,  then 
the  last  of  the  above  equations  will  be  sufficient  to  establish  the 
equilibrium,  the  fixed  point  being  taken  for  the  centre  of  moments. 

If  of  the  equations  (1)  only  the  last  has  place,  then  all  we  -can 
infer  is  that  the  system  has  no  tendency  to  move  round  the  point 
taken  for  the  centre  of  moments,  and  since  we  here  suppose  that 
the  value  of  R,  viz.  R=v^{X^-f  Y^,5  is  something,  but  that  Rr  or 
2  (P/j)  is  0,  we  must  conclude  that  r==0,  so  that  the  point  taken 
for  the  centre  of  moments  must  be  on  the  resultant,  in  order  that 
the  equation  S  (P/;)=0  may  have  place,  and  for  every  point  on 
the  resultant  it  obviously  will  have  place ;  so  that  if  any  point  on 
the  resultant  were  to  be  fixed,  this  point  would  sustain  the  whole 
pressure  of  the  system. 

//.  TVhen  the  forces  are  situated  in  different  planes. 

(55.)  Let  us  now  consider  the  forces  P,  P^,  P^,  &c.  as  acting,  in 
any  manner  whatever,  upon  a  solid  body,  or  upon  a  system  of  points, 
invariably  connected.  Let  the  inclinations  of  P  to  the  axes  of 
X,  y,  and  z,  be  as  usual  a,  (3,  y ;  the  inclinations  of  P^,  a^,  |3j,  y^ ; 
and  so  on.  Moreover  let  the  co-ordinates  of  any  point  in  P's  di- 
G  2 


78  ELEMENTS  OF  STATICS. 

rcction  be  .r,  y,  and  z  ;  those  of  any  point  in  P,'s  direction,  a-j,  y^, 
2^,  &c  ;  and  let  all  these  directions  be  produced  till  thev  pierce  one 
of  the  co-ordinate  planes,  say  the  plane  of  xy,  excluding  for  the 
present  those  forces  which  may  be  parallel  to  this  plane.  Let  us 
call  the  co-ordinates  of  the  point  where  P  pierces  the  plane  of  xy, 
x\  and  y',  then,  for  the  equations  of  this  line,  we  shall  have, 
{Anal.  Geom.,  p.  225-6,) 

y—y-=bzS  ''•■'<}' 
where  a  and  b  denote  the  tangents  of  the  angles  which  the  projec- 
tions of  the  proposed  line  on  the  vertical  planes  make  with  the 
axis  of  z.  Now  between  these  constants  a,  b,  and  the  given  incli- 
nations of  the  line  itself  to  the  axes,  there  exists  the  relations, 
{Anal.  Geom.  p.  228,) 

cos.  a=a  cos.  y  ,  COS.  /3=i6  cos.  y 
cos.  a     ,       COS.  (3 

.-.  a= ,  b— ; 

COS. y  COS.  y 

hence  substituting  these  values  in  the  equations  (1),  and  putting  z 
=0,  we  have,  for  the  co-ordinates,  x',  y',  of  the  point  where  P 
pierces  the  plane  of  xy,  the  values 

,       X  COS.  y ZCOS.  a       a;  P  COS.  y 2  P  cos.  a 


cos.  y  P  COS.  y 


(2) 


y  COS.  y 2  COS.  3  ■»/ P  COS.  y 2  P  COS.  j3 

w'=" ^- .  .  . .  (3j  ; 

"'  COS.  y  P  COS.  y  ^    ^ 

at  this  point  (x',  y' ),  since  it  is  in  P's  direction,  we  may  consider 
P  to  be  applied ;  let  us  then  do  so  and  decompose  P  thus  applied, 
according  to  the  three  axes,  and  we  shall  then  have,  instead  of  the 
original  force  P,  these  three,  viz.  P  cos.  a,  P  cos.  /3,  P  cos.  y  ;  the 
first  two  acting  in  the  plane  of  xy,  and  the  third  acting  parallel  to 
the  axis  of  2,  and  all  upon  the  point  (x',  y' )  where  P's  original 
direction  pierces  the  horizontal  plane.  Hence  the  moments  of  the 
vertical  force  P  cos.  y  with  respect  to  the  vertical  planes,  are,  by 
taking  account  of  equations  (2)  and  (3)  just  given, 

X'  P  COS.  y  =  P  COS.  y 2  P  COS.  o 

y'  P  COS.  y=.y  P  COS.  y  —  2  P  cos.  (3. 
Now  whatever  we  have  said  respecting  the  force  P  and  its  com- 
ponents, equally  applies  to  any  other  force  P  „,  that  is,  it  is  equiva- 
lent to  the  three  component  forces  P„  cos.  a„,  P^cos.  3  ^,  P^  oos.y^  ; 
acting  on  the  point  where  P  „'s  direction  pierces  the  horizontal 
plane :  the  first  two  being  in  that  plane  and  parallel  to  the  axis  of 
X  and  y,  and  the  third  force  being  vertical,  or  parallel  to  the  axis  of 
2 ;  moreover  the  moments  of  this  last  force,  in  reference  to  the 
vertical  planes,  must  be 

a^«  P„  t'o^-  r„  —  ^™  P„  <^os.  a,  and  j/„  P„  cos.  y„  —  2„  P„  cos.  f3„  ; 


EQUILIBRIUM    OF    A    SOLID    BODY. 


79 


C   «  ti   m   « 
2  -C     ■    -   - 


s*  g  S 


IC     (M    <D     o 

o   5^  ;::3   n   c 


■O  *i 


3  s-C  «    n    « 
cr  m    o    rt  -1?^.^ 

52    «    o  J3  -S  "^  c3 


S   2 


o  o  "^ 


« si  ^^  § 

c  —  -^  CL,  I  .  2 


<u  -  a"  s   c  -^  ~   72 

o               >~.  OJ  3    cj    u 

rx.^—  0)    (U     -H  O    t)     ^ 

S  oj  ~   ..  I-"   o  '■-" 

<u  i-  ^  ■;:;;  ,o  =«  t„    '~^ 

<»  „,         "  ^  •-'  t_ 

^  2  X^    cu  «    K    o 


rt    O     rt  ^ 


(L>    S 


03     a; 


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6K  ^. 


o 

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80  ELEMENTS    OF    STATICS. 

not  concur,  the  body  would  not  rquilibrato,  but  woxild  have  a  ten- 
dency to  revolve  round  a  fixed  point  in  the  middle  of  the  line, 
joining  the  points  of  application  of  the  two  resultants.  As,  how- 
ever, the  equation  under  consideration,  the  first  of  (6),  thus  far  fixes 
a  point  in  the  system,  it  prevents  the  body  from  tending  to  move  in 
the  direction  of  the  axis  of  x.  Applying  the  same  reasoning  to  each 
of  the  other  two  equations  of  the  group  (6),  we  infer  that  they  are 
sufficient  to  prevent  in  the  body  all  tendency  to  motion  in  the  direc- 
tion of  the  axes  of  y  and  z,  so  that  the  entire  system  cannot  tend  to 
change  its  place,  that  is,  there  can  be  no  motion  of  translation. 

There  may,  however,  exist,  in  conjunction  with  these  conditions, 
a  tendency  to  rotation  in  three  distinct  directions,  and  these  three 
tendencies  must  be  counteracted,  before  the  equilibrium  of  the  body 
can  be  established.  It  must  be  these  tendencies  then  that  are  coun- 
teracted by  the  remaining  three  conditions  (7) ;  but  let  us  examine 
into  the  meaning  of  these  equations,  arid,  for  this  purpose,  it  will 
be  sufficient  to  consider  the  first  term  y  P  cos.  a —  x  P  cos.  j3.  The 
forces  P  COS.  a,  P  cos.  f3,  act  perpendicularly  to  each  other  on  the 
point  to  which  P  is  original^-  applied,  and  in  a  plane  perpendicular 
10  the  axis  of  z ;  y  is  the  distance  of  that  point,  in  the  axis  of  2, 
which  is  in  this  plane,  from  the  direction  of  the  force  P  cos.  a,  and 
X  is  the  distance  of  the  same  point  from  the  direction  of  the  force 
P  cos.  ,3  ;  so  that  if  these  distances  were  rigid  lines,  these  two  forces 
would  tend  to  turn  the  system  about  this  point,  that  is  about  the  axis 
of  z  in  contrary  directions  ;  hence  the  expressions  y  P  cos.  a  and 
X  P  COS.  j3  are  the  moments  of  the  forces  P  cos.  a  and  P  cos.  ;3,  with 
respect  to  the  axis  of  z,  or,  which  is  the  same  thing,  with  respect 
to  the  point  where  their  plane  cuts  the  axis  of  z.  Hence  the 
gi'oup  of  equations  (7)  intimate  that  in  the  ease  of  equilibrium, 
the  sicm  of  the  moments  of  the  component  forces,  ivith  respect  to 
the  three  axes,  are  severally  0  ;  understanding  by  the  moment  of  a 
force,  with  respect  to  an  axis,  the  product  of  that  force  by  the  dis- 
tance of  the  axis  from  its  direction,  the  force  itself  always  acting  in 
a  plane  perpendicular  to  the  axis.  AVe  have  thus  a  right  to  infer, 
seeing  that  equations  (6)  and  (7)  fully  establish  the  equilibrium, 
that  when  there  is  no  motion  of  translation,  and  when  moreover  the 
moments  of  rotation  about  three  rectangular  axes  are  each  0,  the 
moments  of  rotation  about  any  other  axes  whatever  must  be  0. 

It  remains  to  consider  the  elfect  of  those  among  the  original  forces 
which  may  be  parallel  to  the  plane  of  xy,  since  the  decomposition 
we  have  hitherto  employed  supposes  the  forces  all  to  meet  this  plane. 
Suppose  P  to  be  parallel  to  the  plane  of  xy ;  at  its  point  of  applica- 
tion apply  also  two  equal  but  opposite  vertical  forces  Q,  —  Q,  then 
one  of  these  being  directed  towards  the  plane  of  xy,  the  component 
of  this  and  P  will  meet  that  plane. 


EQUILIBRIUM  OF  A  SOLID  BODY.  81 

Call  this  resultant  P',  then,  instead  of  the  force  P,  we  shall  have 
the  two  forces  P'  and  Q  acting  at  the  same  point,  and  to  which  the 
preceding  theory  is  applicable.  Now  the  horizontal  components  of 
P'  are  the  same  as  the  horizontal  components  of  P,  seeing  that  P' 
is  composed  of  P  and  a  vertical  force ;  and  therefore  the  two  first 
of  equations  (6)  and  the  first  of  (7)  remain  unaltered,  whether  we 
substitute  the  two  forces  P'  and  Q  for  P  or  not.  For  the  third  equa- 
tion of  (6),  these  two  forces  furnish  the  terms  P'  cos.  y+Q= — Q 
-}-Q=0,  the  same  as  the  original  force  P  for  which  cos.  y  is  0. 
For  the  second  of  equations  (7),  these  same  forces  furnish  the  terms 
(x  P'  cos.  y  —  zP'  COS.  a)=( —  xQ  —  z  P  COS.  a)  and  (x  Q,  —  0), 
the  aggregate  of  which  is  the  terra  —  z  P  cos.  a,  the  same  that  would 
be  given  by  the  original  force  P,  for  which  cos.  y  is  0  ;  and  pre- 
cisely in  the  same  way  is  it  to  be  shown,  that  there  will  be  no  dif- 
ference in  the  last  equation  whether  we  substitute  for  P  the  forces 
P',  Q,  or  take  the  force  P  itself.  Hence  the  foregoing  equations 
establish  the  equilibrium,  however  the  original  forces  may  act  on 
the  system. 

(57.)  When  the  body  to  which  the  forces  are  applied  is  not  free, 
but  is  only  at  liberty  to  move  in  any  direction  round  a  fixed  point, 
then,  taking  this  point  for  the  origin,  the  equilibrium  will  be  esta- 
blished if  the  equations  (7)  have  place,  for  these  equations  forbid  all 
tendency  to  move  in  the  only  way  in  which  the  body  is  at  liberty 
to  move. 

In  such  a  case  the  pressure  on  the  point  must  be  equal  in  inten- 
sity to  the  resultant  of  all  the  applied  forces,  or  to  the  resultant  of 
the  same  forces  if  they  were  all  to  be  transferred  parallel  to  their 
directions  to  the  fixed  point  itself,  since,  as  is  plain  from  equations 
(6),  it  will  require  the  same  pressure  or  opposing  force  to  preserve 
the  equilibrium  in  one  case  as  in  the  other.  Calling  the  pressures 
on  the  point,  estimated  in  the  several  directions  of  the  axes  of  x,y, 
and  z,  X,  Y,  and  Z,  we  have,  from  equations  (6), 

X= 2  (P  COS.  a),  Y= S  (P  COS.  |3),   Z=: S   (P  cos.  y). 

When  the  body  can  move  only  round  a  fixed  axis,  then,  taking 
this  for  the  axis  of  z,  all  tendency  to  this  motion  will  be  prevented 
if  the  single  equation 

(y  P  cos.  a — X  Pcos.  p)  +  (2/i  Pi  cos.  a^  — .r,  Pj  cos.sj-f 
(2/2  P2  COS.  ttg  ' —  X2  P2  COS.  |32)-}-&c.^0,  exist. 

We  may  consider  the  fixed  axis  to  be  secured  at  its  extremities 
by  two  fixed  points  or  pivots,  and  then  the  general  equations  of 
equilibrium  will  furnish  us  with  expressions  for  the  pressures  sus- 
tained by  these  pivots.  Call  the  pressures  upon  one  of  the  points, 
(0,  0,  Z',)  in  the  directions  of  the  axes,  X',  Y',  Z' ;  and  the 
pressures  upon  the  other  point,  (0,  0,  Z",)  X",  Y",  Z"  ;  then, 
these  being  the  forces  which,  with  those  applied  to  the  body,  secure 

11 


82  ELEMENTS  OF  STATICS. 

the  equilibrium  of  the  system,  we  must  liavc,  between  them  and 
the  applied  forces,  the  conditions 

X'  +  X"=—  2  (P  COS.  a),Y'  + Y"=—  2  (P  COS.  ,3),  and 
Z'  +  Z"= —  2  (P  COS.  y) .  (1),  in  order  that  the  equations  (6)  may  be 
satisfied  ;  and  likewise  the  conditions 

Z'X'  +  Z"  X"=—l:  (xF  COS.  y  —  zT  COS.  a)   I  ,„x 

z'  Y"+z"  Y"=  —  2  Cy  P cos. y  —  r  P cos.  ,i)S""^  ^' 
in  order  that  the  equations  (7)  may  be  satisfied. 

The  third  of  the  equations  (1)  shows  that  the  axis  is  pressed  in 
the  direction  of  its  length  by  the  force  2  (P  cos.  y) ;  which  is,  there- 
fore, divided  between  the  two  pivots,  although  there  is  nothing  to 
teach  us  in  what  proportion.  The  remaining  four  equations  of  con- 
dition enable  us  to  determine,  when  the  intensities  and  directions, 
as  well  as  the  points  of  application  of  the  applied  forces,  are  given, 
what  efforts  they  exercise  on  each  pivot  to  carry  away  the  axis. 

(58.)  Before  concluding  the  present  chapter,  we  shall  briefly 
advert  to  the  remarkable  analogy  which  exists  between  the  theory 
of  moments  and  the  projections  of  plane  figures  in  geometry;  and 
first  we  may  remark,  that  the  several  terms  which  constitute  the 
equations  (7),  and  each  of  which  we  have  called  the  moments  of 
two  component  forces  with  respect  to  the  axis  perpendicular  to  their 
plane,  may  each  be  considered  as  the  moment  of  the  projection  of 
the  force  itself  on  the  plane  perpendicular  to  the  same  axis,  Avith 
respect  to  the  origin  A.  Thus  let  us  take  any  force  P„,  and  repre- 
sent it  by  a  part  of  its  direction  estimated  from  the  point  of  appli- 
cation ;  then  the  projection  of  the  line  P„  on  the  plane  of  xy  may 
be  considered  as  the  projection  on  that  plane  of  the  proposed  force. 
It  is  obvious  that  the  components  of  this  projection  parallel  to  the 
axes  of  X  and  y  must  be  the  very  same  as  the  components  of  P^ 
parallel  to  these  axes,  that  is,  this  projection  is  the  resultant  of  the 
two  forces  P,  cos.  a„  and  P„  cos.  j5„  ;  but  when  two  forces  act  in  a 

n  -  n  n  '  n  ' 

plane,  the  moment  of  the  resultant  in  reference  to  a  fixed  point  is 
equal  to  the  sum  of  the  moments  of  the  components,  or  to  their  dif- 
ference if  they  tend  to  turn  the  system  in  contrary  directions  about 
this  point,  and  the  moment  of  our  two  forces  is  y^  P^  cos.  o„,  and 
x^  P„  COS.  /3„.  Hence,  calling  the  resultant  of  these  forces,  or  the 
projection  of  P„,  P'^  ;  and;;',,  the  perpendicular  upon  it  from  the 
centre  of  moments,  A,  we  have 

/''n  P'n=3/n  P;"  COS.  a„  —  X„  P„  cos.  |3„  , 

and  similar  expressions  obviously  have  place  for  the  moments  of 
the  projections  on  the  other  two  planes. 

As  the  projection  of  a  line  on  a  plane  is  the  product  of  the  line 
by  the  cosine  of  its  inclination  to  the  plane,  or  by  the  sine  of  its 
inclination  to  a  perpendicular  to  the  plane,  the  projection  P'„,  of 
J'„  on  the  plane  of  xy,  will  be  expressed  by  the  product  P^  sin. y„; 


EQUILIBRIUM    OF    A    SOLID    BODY.  83 

and,  in  like  manner,  the  projections  P",,  and  P"'„,  of  the  same 
line  on  the  planes  xz  and  yz  will  be  P"„  =P„  sin.  /3„,  P"'„  = 
P  sin.  a„  ;  hence,  calling  the  perpendiculars  on  these  projections 
from  the  origin,  or  centre  of  moments,  'p'\  and  ^9'",^,  we  may  write 
the  equations  (7)  in  this  more  concise  form,  viz. 
/J'    Psin.7+/)',    P,sin.yj+7/3    P^sin.  y^+&c.  =  01 

p"  P  sin.  i3+p",  PiSin./3j-f/>"3  P^sin.  |3^-f  &c.=0  K (7). 

/j"'Psin.  a4-7)"\  P^sin.aj+jt;"'2PgSin.  a3  +  &c.=0  j 

If /),j  be  the  perpendicular  from  the  origin,  or  centre  of  moments, 
upon  the  line  P„  in  space,  then  /)„  P,^  will  be  double  the  area  of  the 
triangle  whose  base  is  P,^  and  vertex  the  origin ;  also  /)'„  P'„  will 
be  double  the  area  of  its  projection  ;  we  may  say,  therefore,  that  the 
moment  of  the  projection  of  any  force,  is  equal  to  the  projection  of 
its  moment. 

As  the  moment  /)'  P'  of  the  projection  of  any  force  P  on  the 
plane  of  xxj,  or  the  moment  jo"  P",  of  the  projection  on  the  plane 
of  rz,  or  the  moment  jo'"  P'",  of  the  projection  on  the  plane  oi yz, 
is  in  each  case  equal  to  double  the  projection  of  the  triangle  whose 
base  is  P  and  vertex  the  origin,  or  centre  of  moments,  it  fol- 
lows, from  the  theory  of  projections,  {Anal.  Geom.  p.  256,)  that  if 
p""  P""  represent  the  moment  of  the  projection  of  the  same  force 
on  any  fourth  plane,  passing  through  the  centre  of  moments,  we 
must  "have  S  {p""  P"")=cos.  5  S  {p'V)-j-cos.  b'  S  {p"  P")  + 
cos.  b"  ^  (p'"  P'"),  where  6,  6',  8",  are  the  inclinations  of  the  new 
plane  to  the  planes  of  j5'  P',  of/j"  P",  and  of  p'"  P'"  respectively. 

It  also  follows  from  the  same  theory,  (Anal.  Geom.  p.  256,)  that 
if  any  number  of  forces  be  projected  on  three  rectangular  planes, 
and  the  moments  of  the  projections  on  each  plane,  regarding  the 
origin  as  the  centre,  be  collected  into  one  sum,  the  squares  of  the 
three  sums  thus  furnished  will  be  constant  for  every  system  of  rect- 
angular planes  having  the  same  origin. 

For  the  determination  of  the  principal  plane,  or  that  in  which 
the  moments  of  projection  of  any  given  forces  amount  to  the  great- 
est sum,  see  the  Analytical  Geometry,  p.  256  et  seq. 


84  ELEMENTS   OF    STATICS. 

CHAPTER  IV. 

PROBLEMS    ON    THE    EQUILIBRIUM    OF    A    SOLID    BODY. 

Problem  I. — (59.)  A  bent  rod  or  lever  ACB  is  suspended  at  C, 
about  which  point  it  is  free  to  move  in  a  vertical  plane,  and  weights 
P,  P3  arc  attached  to  its  extremities :  to  find  the  position  in  which 
it  will  rest  (fig.  47). 

Let  us  take  the  fixed  point  C  for  the  centre  of  moments,  then  it 
will  be  sufficient  for  the  equilibrium  that  the  moments  of  the  applied 
forces  balance  each  other.  These  applied  forces  are  first,  the  weight 
Pj  of  the  arm  CA  acting  at  its  centre  of  gravity,  the  middle  point 
a  ;  secondly,  the  weight  P  acting  at  A  ;  thirdly,  the  weight  P^  of 
the  arm  CB  acting  at  the  middle  point  b  ;  and  lastly,  the  M'eightPg 
acting  at  B  :  these  forces  all  have  vertical  directions,  and  the  mo- 
ments of  the  two  latter  oppose  the  moments  of  the  two  former. 
Hence,  drawing  the  perpendiculars  from  C  on  the  directions  of  the 
forces  as  in  the  figure,  we  have  this  equation  for  the  conditions  of 
equilibrium,  viz. 

P  .  C;)+P, .  C;;,=Pj .  Qp,+y, .  Cp,, 
and  it  is  from  this  equation  that  the  required  position,  that  is,  the 
angle  ACp  or  the  angle  BCpg,  must  be  determined.     Put  o  for  the 
angle  AC;;,  a  for  the  angle  BC/)^  and  e  for  the  given  angle  ACB ; 
also,  call  the  given  length  AC,  2  o,  and  the  given  length  BC,  2  a',  then 

Cj9=2a  COS.  a,  C/;j=a  cos.  o,  Cp^=a'  cos.  a'=a'  cos.  (a-f  6)> 
C/>3=2rt'  COS.  a'= — 2a'  cos.  (a-fe); 
hence  the  equation  of  equilibrium  is  the  same  as 

(2P  +  P,)  a  COS.  a=— (P„+2P3)  a'  cos.  (a  +  e), 
where  a  is  the  only  unknown  quantity.     Or,  since 

cos.  (a+e)       sin.  o  sin.  6  —  cos.  o  cos.  9 

^^ = =  tan.  asm.  9 — cos.  9, 

cos.   a  cos.  a 

the  equation  reduces  to 

(2P  +  PJ  a=P3+2P3)  «'  (tan.  o  sin.  e — cos.  e) 
(2P-f-PJ«+(P.+2P3)a'cos.e 

••    ^"•'*  (P,+2P3)  a' sin.  0 

If  tlie  extremities  of  the  lever  carry  no  weights 

P.  a  +  P„  a'  COS.  9 

tan.  a= p; — ^. . 

PjjO  sin.  9 

These  results  would  remain  unaltered,  though  the  two  arms  CA, 

CB  were  of  difierent  thickness  ;  but  when  they  are  equally  thick, 

P         a' 
since  their  weights  must  then  be  as  their  lengths,  or,  since  -^^  =  — , 


EftDILIBRlUM    OF    A    SOLID    BODY.  8^ 

the  last  expression  may  be  put  under  the  form 

ff^+a'^  COS.  e 

tan.  a= — ^7— ^ . 

a^  sm.  d 

Problkm  II. — (60.)  An  oblique  cylinder  stands  on  a  horizontal 
plane,  its  inclination  to  which  is  60°,  perpendicular  height  4  feet, 
and  diameter  of  the  base  3  feet.  Required  the  diameter  of  the 
greatest  sphere  of  the  same  material  as  the  cylinder  that  will  hang 
suspended  from  the  upper  edge  (fig.  48,)  without  overturning  the 
cylinder. 

The  centre  of  gravity  G  of  the  cylinder  is  at  the  middle  of  its 
axis  CD  ;  hence  the  acting  forces  are  the  weight  of  the  cylinder  in 
the  vertical  direction  GE,  and  the  weight  of  the  sphere  in  the  ver- 
tical direction  B'P  and  these  two  forces  are  just  sufficient  to  prevent 
any  tendency  to  motion  about  B  ;  hence  the  equation  of  equilibrium  is 

BE  X  cylinder=BF  x  sphere. 

Now  by  trigonometry  sin.  GDE  :  DGE  :  :  GE  :  DE=2v/5 

.-.  BF=2DE=4v^|,  BE=BD  — DE=|  — 4^i  ; 

hence,  substituting  the  volumes  of  the  cylinder  and  sphere  for  their 

weights  to  which  they  are  proportional,  the  equation  of  equilibrium  is 

(I— 4v/5)x32x  -7854 x4=4^^x§X  -7854 xdiameter  1  ^    ^ 
.-.  diameter=2  •  006  feet. 

Problem  III. — (61.)  One  extremity  C  of  a  heavy  rod  is  move- 
able about  a  fixed  point  in  a  vertical  plane  (fig.  49),  and  to  the  other 
extremity  B  is  fastened  a  cord  which  goes  over  a  pulley  A  in  a  ho- 
rizontal line  with  C,  and  supports  a  weight  P  equal  to  half  the 
weight  of  the  rod  :  required  the  position  in  which  the  rod  will  rest. 

The  forces  which  prevent  motion  about  C  are  the  tension  of  the 
cord  BA,  measured  by  the  weight  P,  and  the  weight  2P  of  the  rod 
acting  vertically  at  the  middle  of  CB. 

Draw  BF  perpendicular  to  AC,  and  CE  perpendicular  to  AB, 
then  the  equation  of  equilibrium  is 

P  .  CE=2P  .  i  CF=P  .  CF  .-.  CE=CF. 
Put  BC=fl,  AC =6,  and  AF=a',  then   the  right-angled  triangles 
AEC,  AFB  being  similar,  we  have 

EC''_BF^  ^  {b—xY_    a^—jb—xY    _  a^~^b^-\-2bx  —  x'' 
AC^~  BA^*"'      ¥~  ~x^+a^—{b  —  xy~       aF^—b^+2bx 

an  equation  which  reduces  to 

2bx^  —  {'ib^—a^)  x^'—'Zb  (a^'—b)  x=0. 

One  root  of  this  equation  is  x=0,  and  the  other  two,  as  given  by  the 

quadratic,  x^ — — x=a^ — b^  .  (1) 

H 


86  ELEMENTS    OF    STATICS. 

are  x=b —^{a±^{Sb^+a^)\ (3). 

Let  us  examine  into  tlie  nature  of  these  as  connected  with  the  posi- 
tions of  the  rod.  Tlie  first  root  .r=0  gives  the  position  in  fig.  50, 
which  position,  however,  the  rod  cannot  take  if  AC  is  greater  than 
CI3,  that  is  if  b  exceeds  a  ;  but  when  b  does  not  exceed  a,  then  this 
will  be  one  of  the  position-s  of  equilibrium,  and  it  may  be  remarked 
that  whatever  be  the  weight  of  the  rod  moveable  about  C,  tlie  other 
end  B  will  always  be  supported  in  every  position  by  a  vertical  force 
equal  to  half  that  weight. 

There  cannot  be  any  other  position  of  equilibrium  for  the  same 
weight  P  between  the  lines  CB,  CA,  because  in  no  such  position  can 
the  condition  of  equilibrium,  viz.  CE^CF,  have  place ;  hence  no  ne- 
gative root  of  (1)  can  be  consistent  with  the  conditions  of  the  question. 

To  determine  in  what  circumstances  the  two  roots  are  admissible, 
put  a=n6,  then  the  values  of  x  will  be 

which  for  ?i  =  l  gives  .r=0,  and  a'=:|6  ;  when  n  is  less  than  1, 
then  it  is  obvious  that  both  values  of  x  are  positive ;  hence,  when 
a  is  not  greater  than  b,  there  are  always  two  positions  of  equilibrium 
determinable  from  the  two  roots  or  values  of  x  in  (2).  As  one 
of  these  values  of  x  is  always  greater  than  b,  the  corresponding  po- 
sition of  the  rod  will  be  as  in  fig.  51.  When  n  is  greater  than  1, 
one  of  the  above  values  of  x  is  always  positive  and  the  other  always 
negative  ;  hence,  when  a  is  greater  than  b,  there  is  but  one  position 
of  equilibrium  (fig.  52)  determinable  from  the  positive  root  or  value 
of  X  in  (2),  but  then  there  is  another  position  determinable  from 
x=0,  that  is,  the  extremity  B  will  rest  in  the  vertical  line  from  the 
pulley,  as  in  fig.  50. 

Problem  IV. — (62.)  AD  and  BC  (fig.  53)  are  two  heavy  bars 
moveable  in  a  vertical  plane  about  their  extremities  A,  B  in  the 
horizontal  line  AB  ;  required  the  position  in  which  they  will  rest 
by  leaning  against  each  other. 

Call  the  weight  of  the  bar  AD,  acting  vertically  at  its  middle 
E,  P  ;  and  the  weight  of  the  bar  BC,  acting  vertically  at  G,  P^ ;  then 
the  forces  acting  on  AD,  to  turn  it  round  A,  are  P  in  the  direction 
EK,  and  the  pressure  of  CB  in  the  direction  DL  perpendicular  to 
BC  ;  call  this  pressure  P^,  then  the  bar  AD  being  at  rest,  we  must 
have  P  .  AK=P3  .  AL,  also  the  forces  acting  on  BC,  to  turn  it  round 
B,  are  Pj  in  the  direction  GH,  and  the  pressure  P^  in  the  perpen- 
dicular direction  LD ;  hence,  Pj.BH=P3.BD,  these  two  are 
therefore  the  equations  of  equilibrium.  Eliminating  P^,  we  have  the 
single  equation 


EaUILlBRlUM  OF  A  SOLID  BODY.  87 


P.AK.BD=P,.BH.AL. 
Put  AB=a,  AD=6,  BC=c,  and  BD=a?;  then 

ATT         KT?  A         hb{b'-^a^—X') 

AK=AE  •  COS.  A= ^^ — —- 

2ab 


BH=BG.cos.B=: 


AL=AD  .  cos.  D  = 


2  ax 


2x 
Hence,  by  substitution,  we  have 

2b{b''+a''—x^)V=bc{a^+x^~b^)  (b^-\.x^  —  a'>)F^; 
an  equation  of  the  fifth  degree,  ffom  which,  when  numbers  are  put 
for  a,  b,  and  c,  the  vahies  of  x  may  be  determined,  (see  Algebra, 
p.  210.) 

Problem  V.— ;-(63.)  A  given  rod  or  beam,  not  of  uniform  thick- 
ness, has  one  end  suspended  by  a  cord  of  a  given  length,  fixed  at 
a  given  point  above  an  inclined  plane  of  a  given  inclination,  and 
the  other  end  of  the  beam  is  sustained  by  the  inclined  plane  ;  it  is 
required  to  determine  the  position  of  the  beam,  weight  sustained 
by  the  cord,  and  pressure  against  the  inclined  plane  when  the  beam 
is  at  rest  (fig.  54). 

Let  P  be  the  given  point,  AP  the  string,  AB  the  position  of  the 
beam  when  at  rest,  with  its  end  B  on  the  given  inclined  plane  BK. 

The  forces  acting  on  the  beam  are  the  tension  of  the  strino-  in 
the  direction  AP,  the  weight  of  the  beam  acting  vertically  at  the 
given  point  G,  its  centre  of  gravity,  and  the  pressure  at  B  acting 
in  the  given  direction  BD',  perpendicular  to  the  plane.  Let  PA, 
D'B,  be  produced  till  they  meet  in  D,  then,  as  the  resultant  of  the 
forces  acting  in  these  lines  must  pass  through  D,  and  as  the  direc- 
tion of  this  resultant  is  vertical,  it  follows  that  the  vertical  line  DG 
must  pass  through  the  centre  of  gravity.  Hence  to  determine  the 
position  of  the  rod,  draw  AH,  PK,  parallel  to  D'D,  and  AM  paral- 
lel to  BK,  also  produce  BA  to  N  ;  then  the  position  will  be  ascer- 
tained, if  we  can  find  the  angle  PCK=PAM=0.  The  known 
quantities  are  BG=fl!,  GA=b,  PK=c,  PA=/,  CBQ=i=inclination 
of  the  plane ;  if,  therefore,  we  call  the  unknown  angle  ABK,  or 
NAM,  ^,  we  shall  have 

PM=;sin.  e,  AH=KM=(a+6)  sin.  p 
.-.  PM  — KM=/  sin.  6-\-{a-\-b)  sin.  p=c  ....  (A). 
If,  therefore,  we  can  obtain  another  equation  involving  no  other 
unknown  quantities  besides  e  and  ^,  this  latter  may  be  eliminated, 
and  thence  e  determined.     Now  two  different  expressions  for  BD, 
involving  only  these  unknown  quantities,  may  readily  be  obtained 


88      ,  ELEMENTS  OF  STATICS. 

from  the  two  triangles  ABD,  GBD,  which  have  this  side  in  com- 
mon ;  for  observing  that 

PAN=BAD=9  — t 

ADB=APM=90°— » 

BDG=KBQ=/ 

BGD=comp.  QBG=90°— (i+t); 

we  have,  from  the  triangle  ABD, 

sin.  ADB  :  sin.  BAD  : :  AB  :  BD  ; 

that  is,         cos.  9  :  sin.  (9  —  ^)  :  :  a+6  :  BD  .   .  .   .  (1)  ; 

and  in  the  triangle  BGD,  we  have 

sin.  GDB  :  sin.  DGB  :  :  BG  :  BD  ; 

that  is,  sin.  i:  cos.  (i+t)  '•  •  a-  BD  ....  (2). 

Equating  now  the  two  expressions  for   BD,  furnished  by  (1)  and 

,„.  ,  (a+b)  sin.  (e  —  *)     o  cos.  (i+*) 

(2),  we  have      l-J—i ^^ lL= —^-^^-^ 

COS.  e  sm.  t 

.'.  (a+6)  sin.  i  sin.  (s  —  ^)=a  cos.  B  cos.  (i+$)  ....  (B) ; 
hence,  by  means  of  equations  (A)  and  (B),  6  may  be  determined, 
and  thus  the  position  of  the  beam  found. 

It  remains  to  determine  the  tension  T  of  the  string,  and  the  pres- 
sure P  on  the  inclined  plane.  In  order  to  this  draw  GV  parallel 
to  PA,  then,  since  the  sides  of  the  triangle  GVD  are  respectively 
parallel  to  the  three  forces,  they,  or  the  sines  of  their  opposite 
angles,  are  proportional  to  these  forces ;  that  is,  calling  the  force  in 
GD,  or  weight  of  the  beam  AV,  we  have 

sin.  GVD  :  sin.  VDG=sin.  z  : :  W  :  T 
sin.  GVD  :  sin.  VGD=cos.  9  :  :  W  :  P  ; 

,  _,     sin.  i  „^    _.     COS.  (94-i)  ^.. 

consequently,      T= W,  P= ^  ^  MV. 

COS.  e  COS.  e 

Problem  VI. — (64.)  A  given  beam  AB  is  supported  by  strings 
which  go  over  pullies  C,  D,  and  have  given  weights  P,  P^,  attach- 
ed to  them,  to  find  the  position  of  equilibrium  (fig.  55). 

Produce  the  strings  till  they  meet  in  g,  then  the  vertical  ^E  will 
pass  through  the  centre  of  gravity  G  of  the  beam,  and  if  GF  be 
drawn  parallel  to  D^  the  three  sides  of  the  triangle  GFg  will  be 
proportional  to  the  three  equilibrating  forces,  and  these  are  all 
given.  Put  AG=fl,  GB=&,  the  inclination  of  CD  to  the  horizon- 
tal line  CK,  or  DK',  i ;  these  quantities  are  also  given.  Let  a 
represent  the  angle  DCA,  and  (3  the  angle  CDg,  then  sin.  BDPj= 
cos.  (/3+z)=sin.  FG^,  and  cos.  ^CK=cos.  (a — i)=sin.  F^G  ; 

P       cos.  (3-\-l^ 

hence,  from  the  triangle  GFe-,  we  have  -r-= '-p :{    ...  (1) 

^  *  Pj      COS.  (tt  —  t)  ^ 

P  _       COS.  (fi+i)        _cos.  (i3-fi) 
W^sin.  (180°— p  — a) ~  sin.  (^+a)' 


EQUILIBRIUM  OF  A  SOLID  BODY.  89 

From  these  two  equations  the  unknown  angles  a  and  /3  may  be 
determined.  But  to  find  the  position  of  AB,  we  must  also  know 
the  angle  A'=S.  From  G  draw  the  perpendiculars  Gp,  Gp^,  on 
the  strings,  then  Gp=GA  sin.  A=a  sin.  (a  —  6),  and 

Gjo,=GB  sin.  GBp^=b  sin.  (f3+6), 
hence  a  sin.  (a — 5)  P=6  sin.  (i3+6)  P^ 

^    F_bsm.(i3  +  6) 

"  P,     a  sin.  (a  — 6)  ^  ^' 

from  which  equation  S  may  be  determined,  a  and  /3  having  been 
previously  found.  The  quantities  thus  found  enable  us  to  find  the 
two  unknown  sides  of  the  triangle  Co-D,  and  those  of  the  triangle 
AgB,  and  thence  the  distances  CA,  DB. 

Problem  VII.— (65.)  A  given  beam  AB  hangs  by  two  strings, 
CA,  DB,  of  given  lengths,  from  two  given  fixed  points  C,  D ;  to 
find  the  position  in  which  it  will  rest  (fig.  56). 

Here  instead  of  the  tensions  we  have  the  lengths  a',  b'  of  the 
strings  C A,  DB ;. hence,  using  the  same  notation  and  the  same 
reasoning  as  in  the  last  problem,  we  get  two  difljerent  expressions 
(1),  (2),  for  the  ratio  of  these  tensions  ;  that  is,  we  have  one  equa- 
tion between  the  unknowns  a,  j3,  and  8 ;  it  will,  therefore,  require 
two  more  equations  to  determine  them  ;  these  two  may  be  readily 
obtained,  since  we  may  deduce  two  different  expressions  for  the 
perpendiculars  from  A  and  B  on  the  strings.  Thus,  if  we  draw 
AM  parallel  to  BD,  it  is  plain  that  the  perpendicular  from  A  on  DB 
will  be  equal  to  that  from  C  on  DB,  minus  that  from  C  on  AM ; 
that  is,  calling  CD,  c,  the  expression  for  this  perpendicular  will  be 

c  sin.  |3  —  a'  sin.  (a+/3) ; 
but  the  expression  for  the  same  perpendicular  is  also  AB  sin.  B, 
that  is  {a+b)  sin.  (i3+S) 

.*.  c  sin.  /3  —  a'  sin.  (a+/3)=(ff+|3)  sin.  (|3  +  8)  ....  (1). 

Again  the  perpendicular  from  B  on  kg  is  equal  to  the  perpendi- 
cular on  it  from  D,  minus  that  from  D  on  BN  parallel  to  A"-,  the 
expression  for  this  perpendicular  is,  therefore, 
c  sin.  a — b'  sin.  (a+/3) ; 
but  the  expression  for  the  same  perpendicular  is  also  AB  sin.  A, 
that  is,  («+6)  sin.  (a — 6) 

.*.  c  sin.  a — b'  sin.  (a+/3)=(o  +  6)  sin.  (a  —  6)  .  .  .  (2). 
These  equations,  in  conjunction  with  that  before  mentioned,  viz. 
with  .cos^+lJ^^sin.O  +  g)  _  ^  ^ 

sm.  (/S+a)     asm.  (a — b)  ^' 

are  sufficient  to  determine  the  unknown  quantities  sought. 

Problem  VIII.— (66.)  A  given  solid  hemisphere,  with  its  convex 
part  upon  a  smooth  inclined  plane  of  given  inclination,  is  kept 
h2  12 


90  ELEMENTS    OF    STATICS. 

from  sliding  by  a  string  of  given  length  having  one  end  fastened  to 
a  given  point,  and  the  other  end  attached  to  the  edge  of  the  hemis- 
phere :  it  is  required  to  determine  the  point  where  the  hemisphere 
touches  the  plane  when  at  rest,  the  pressure  on  the  plane,  and  the 
tension  of  the  string  (fig.  57). 

Let  P  be  the  given  point,  PI  the  string,  IFM  the  hemisphere,  C 
its  centre,  and  CT  perpendicular  to  MI,  then  the  distance  CG  of 
its  centre  of  gravity  is  i  CT.  The  forces  acting  on  the  body  are 
the  weight  W  in  the  vertical  direction  GR,  the  pressure  P  in  the 
direction  FO  perpendicular  to  the  inclined  plane  AB,  and  the  ten- 
sion T  in  the  direction  IP  :  this  last  direction  must  when  produced 
meet  in  O  the  point  of  concurrence  of  the  other  two  forces.  To 
determine  the  point  F,  draw  GN,  IL,  and  PE,  perpendicular  to 
CF,  ID  perpendicular  to  PE,  and  PH  to  AB,  then  the  given  quan- 
titles  3,re 

CG=a,  CF=b,  PH=EF=c,  PI--=/,  cot.  angle  BAR= 

cot.  angle  GON=? ; 

and  we  wish  to  find  HF  or  PD  and  IL.     Put  x  andy  for  the  sine 

and  cosine  of  the  angle  GCN,  or  CIL,  then  we  have 

CN=ff.y,  GN=a.r,  CL=bx,  lA=by 

.'.  0N=/  •  GN=/rt.r,  CO=CN  — ON=fl  (y  —  tx) 

LO=CL'— CO =bx  — a  (y  —  tx),  LE=CE  — CL= 

lY)=b —  c  — bx.     By  the  similar  triangles  OLI,  IDP, 

OL  :  LI : :  ID  :  DP  .-.  DP=M^I^iZlM  ; 

ox  —  a  (y  —  tx) 
and  since  DP+DP2=IP''=  I" 

...(6_c_6x)«+s'-^i^^=^l«=P. 
^  ^       '  o.r  —  a  (y  —  tx) ' 

This  equation  joined  lo  x'^-^-y"  =1  is  sufficient  to  determine  a; 
and  y,  and  thence  DP  and  LI.  The  sides  LI,  LO,  being  now 
known,  the  angle  LOI  is  known ;  hence,  taking  the  centre  of  mo- 
ments at  L  we  have 

OLsin.  LOIxT=OLsin.  LOG  •  W  ;  andLOG=Z.A, 

...  T=  -i'"'  ^^  W  ;  also,  P=T  cos.  LOI+W  cos.  LOG 
sm.  LOI 

=  (sin.  LOG  •  cot.  LOU- cos.  LOG)  W. 

Problem  IX. — (67.)  To  determine  the  position  in  which  a  para- 
boloid ABC  will  rest  upon  a  horizontal  plane  (fig.  58). 

Suppose  P  to  be  the  point  on  which  it  rests,  then  the  pressure  be- 
ing in  the  vertical  direction  PG,  is  in  the  normal  to  the  surface  at 
P,  and,  moreover,  passes  through  the  centre  of  gravity  G.  Taking 
the  axis  AX,  AY  the  equation  of  the  vertical  section,  through  AX 

and  PG  is  y^=2px ;  therefore,  the  subnormal  NG  is  NG=y  ^  = 


EQUILIBRIUM    OF    A    SOLID    EODV.  91 

p  ;  but,  calling  the  altitude  AX  of  the  paraboloid  a,  the  distance  AG 

2  2 

is  (ex.  3,  p.  71)  AG=fa,  .•.  x+p  =—  a  .-.  x=  —  a  — p  ; 

o  o 

this  therefore  is  the  abscissa  of  the  point  on  which  the  body  rests, 

hence,  the  tangent  of  the  inclination  XRP  of  the  axis  to  the  hori- 

dv 
zon,  that  is,  the  value  of -^,  for  the  point  P  is 

tan.e=g=^/A=^|^^3(|«_;,)|(l). 

Besides  the  point  thus  determined,  there  is  no  otlier  on  which  we 
can  place  the  body  where  the  normal  shall  be  equal  to  the  distance 
of  the  centre  of  gravity  from  the  horizontal  plane,  which  equality 
must  exist  in  order  that  the  body  may  rest  on  that  point,  making, 
however,  one  exception,  viz.  when  the  point  is  the  vertex  A  ;  for 
at  tliis  point  the  normal  being  in  the  line  of  direction  of  the  centre 
of  gravity,  the  body  would  necessaril}^  rest  on  it,  be  the  length  of 
the  normal  what  it  may.  In  order  that  the  body  may  be  unable  to 
rest  on  any  other  point  besides  this,  the  altitude  must  be  such  that 
between  the  vertex  and  base  there  shall  be  no  point  whose  noi'mal 
shall  equal  the  distance  of  the  centre  of  gravity  from  the  plane, 
therefore  the  condition  is  that  the  result  (1)  may  be  either  impossi- 
ble or  infinite,  which  requires  that 

3  3 

«  <-2-;'' or  a=— ;) ; 

when,  therefore,  a=^p  we  see  that  the  vertex  is  the  point  at  which 
the  normal  measures  its  distance  from  the  centre  of  gravity.  Hence 
if  any  segment  of  a  paraboloid,  whose  altitude  does  not  exceed  ^p, 
be  placed  any  how  on  a  horizontal  plane,  except  indeed  on  its  base, 
it  must  always  restore  itself  to  an  upright  position  and  rest,  when  it 
does  rest,  on  its  vertex. 

Problem  X. — (68.)  To  determine  the  pressures  exerted  by  a 
door  on  its  hinges  or  on  the  two  pivots  upon  which  it  hangs. 

Call  the  weight  of  the  door  acting  at  its  centre  of  gravity  P  the 
distance  of  the  centre  of  gravity,  that  is,  of  the  direction  of  the  force 
from  the  vertical  axis  x,  and  the  distance  between  the  pivots  z' — z"; 
then  there  being  no  pressures  in  the  direction  of  the  axis  of  y,  the 
equations  1  at  page  81,  become,  in  this  case, 

X'  +  X"=  — PCOS.  a=0,  Z'  +  Z"=  —  P  cos.  y=— P (1) 

.-.  X'=— X"; 
hence  the  equations  (2)  furnish  only  (z'  —  z")  X'=  —  Fx 
Pr  Vr  ' 

'•'  X'=      ,  ,  .-.  X"=  -——T'  .  .  .  (2). 

z  '  — z'  z  — z  ' 

The  equation  (1)  shows  that  the  door  presses  with  its  whole  weight 


92  ELEMENTS   OF    STATICS. 

P,  the  vertical  line  of  the  hinges  ;  and  the  equations  (2)  express  the 
horizontal  forces  on  the  hinges,  the  lower  hinge  being  pushed  in- 
wards, and  the  upper  hinge  drawn  outwards,  each  with  a  force  equal  to 
Px 


Problem  XI. — (69.)  One  end  of  a  uniform  beam  of  weight  W 
is  moveable  round  a  fixed  point  in  a  vertical  plane,  and  to  the  other 
end  is  attached  a  string  which  passes  over  a  pulley,  and  is  loaded 
with  a  given  weight  P  ;  the  fixed  point  and  pulley  are  in  a  horizon- 
tal line,  and  tlieir  distance  asunder  is  equal  to  the  length  of  the 
beam,  which,  however,  is  not  given ;  required  the  angle  0  between 
the  beam  and  horizontal  line  when  the  beam  is  at  rest, 

P  P^  1 

cos..=--±(-— +  -)i. 

Problem  XII. — (70.)  A  cone  of  marble,  the  axis  of  which  is 
twenty  feet,  and  base  diameter,  six  feet,  stands  on  the  edge  of  its 
base,  the  axis  making  an  angle  of  00°  with  the  plane  of  the  horizon  ; 
what  must  be  the  direction  and  intensity  of  the  least  force  applied 
to  its  vertex  that  will  just  sustain  the  cone  in  that  position  ;  the 
weight  of  the  cone  being  284  cwt.  1 
'J  he  least  force  is  !•  377  cwt.  acting  perpendicular  to  the  lower  side  of  the  cone. 

Problem  XIII. — (71.)  What  will  be  the  height  of  the  greatest 
segment  that  can  be  cut  off  a  prolate  spheroid  whose  longer  axis  is 
double  the  shorter,  by  a  plane  perpendicular  to  the  longer  axis,  so 
that  it  may  be  unable  to  rest  upon  any  point  of  its  convex  surface 
except  the  vertex.  ? 

Height  :=  semi,  trans.  X  (3  —  \/5). 

Problem  XIV. — (72.)  A  beam  of  given  weight  W,  rests  with  one 
end  against  a  vertical  wall  and  the  other  upon  an  inclined  plane ; 
calling  the  inclination  of  the  beam  to  the  horizon  i,  the  pressure 
against  the  wall  P,  and  the  thrust  against  the  inclined  plane  T,  to 
determine  the  intensities  of  P  and  T. 

p^      W     .^^v/hin.«z-fjcos.''z^y^ 
2  tan.  V  sin.  i. 

Problem  XV. — (73)  A  ladder  of  uniform  thickness  and  weight 
w  pounds,  is  placed  against  a  vertical  wall,  and  at  a  given  inclina- 
tion i  to  the  horizon,  a  person  weighing  W  pounds  ascends  the  lad- 
der, and  it  is  required  to  determine  the  pressure  at  the  top  and  the 
thrust  at  the  bottom  of  the  ladder  when  the  person  arrives  at  any 
given  height. 


ON  THE  MECHANICAL   POWERS.  93 


CHAPTER  IV. 

ON  THE  MECHANICAL  POWERS. 

(74.)  Having  now  considered  pretty  fully  the  theory  of  the  equi- 
Jibrium  of  forces,  applied  to  different  points  of  a  solid  body,  it  is 
proper  that  we  should  speak  of  those  cases  in  which  the  forces  are 
not  immediately  applied  to  the  body,  on  which  their  influence  is  ul- 
timately exerted,  but  to  some  intermediate  body  contrived  for  the 
purpose  of  transmitting  this  influence  in  the  most  advantageous 
manner.  These  contrivances  are  called  Machines,  and  the  simple 
elements  or  constituent  parts  of  all  machinery  are  called  the  mecha- 
nical povjers.  These  are  six  in  number,  and  are  as  follow  :  the  Le- 
ver, the  TVlieel  and  axle,  the  Pulley,  the  Inclined  Plane,  the  Screw, 
and  the  Wedge. 

The  Lever 

(75.)  A  lever  is  a  rigid  bar  or  rod,  moveable  about  a  fixed  point 
or  fidcrum,  and  it  is  divided  into  three  different  kinds,  depending 
on  the  position  of  this  fulcrum  with  respect  to  the  applied  force,  and 
the  body  to  be  influenced  by  it ;  thus,  if  the  fulcrum  be  between 
the  force  or  power  and  the  body  or  weight,  as  in  fig.  59,  the  lever 
is  said  to  be  of  the  first  kind ;  if  it  be  situated  so  that  the  weight  is 
on  the  middle,  as  in  fig.  60,  the  lever  is  of  the  second  kind,  and 
when,  as  in  fig.  61,  the  power  is  in  the  middle,  the  lever  is  of  the 
third  kind.  In  the  figures  the  rods  are  straight,  but  they  would  still 
be  levers  if  bent  or  curved. 

That  a  lever  acted  upon  by  any  forces  may  be  in  a  state  of  equili- 
brium, it  is,  obviously,  merely  necessary  that  the  sum  of  the  mo- 
ments, taking  the  fulcrum  as  the  centre,  be  equal  to  0  ;  this  condi- 
tion, therefore,  comprises  the  whole  theory  of  the  lever,  so  that 
when,  as  is  commonly  the  case,  the  equilibrating  forces  are  a  weight 

W       p 

W  and  a  power  P,  then  the  condition  is  W p^=Vp  •'•-p-  =—  (0  ; 

r  p^ 

hence  the  weight  and  power  are  to  each  other,  reciprocally,  as  their 
distances  from  the  fulcrum.  If,  as  is  usually  the  case,  the  object  be 
to  balance  W  with  the  least  power  possible,  this  must  act  so  that  jo 
may  be  the  greatest  possible,  and,  therefore,  the  power  must  act  per- 
pendicularly to  the  lever. 

When  the  weight  w  of  the  lever  itself  is  to  be  taken  into  con- 
sideration we  must  view  it  as  a  third  force  acting  at  its  centre  of  gra- 


94  ELEMENTS  OF  STATICS, 

vity  ;  if  we  call  the  distance  of  this  new  force  from  the  fulcrum  g, 
then  the  ahove  equation  will  be  W p^:izH'g=Fp  .  (2);  the  upper 
or  lower  sign  being  used  according  as  this  force  tends  to  favour  or 
to  oppose  W. 

(76.)  The  mechanical  advantage  of  the  single  lever  may  be  con- 
siderably increased  by  combining  several  together,  so  that  the  power 
of  one  may  be  communicated  to  another. 

Thus  in  the  system  of  levers,  represented  in  fig.  62,  if  we  call 
the  arms  which  are  on  the  same  side  of  the  fulcra  as  the  power  p, 
Pi,p.^,  «fec.  and  the  other  arms  p',p\,  p\,  &c.  and,  moreover,  call  the 
powers  acting  at  the  extremities  of  the  latter  arms  Pj,  P^,  &,c.  then 
in  the  case  of  equilibrium  we  shall  have  the  equations 
P;;=P,/;',  Pj>,=Pj^',  l\p^=PsP2'^  <^c. 
and  multiplying  these  together, 

Fpp,p^""=l\p'p\p'^-"; 
or,  the  last  power  P„  being  the  weight  W, 

PpPiP2"-=^P'P\P\'—-^ 
so  that  the  power  is  to  the  weight,  as  the  product  of  tlie  arms  on 
the  side  of  the  weight,  to  the  product  of  the  arms  on  the  side  of  the 
power. 

Problem  I. — (77.)  A  body  is  weighed  successively  in  the  two 
scales  of  a  false  balance,  in  the  one  scale  it  balances  a  weight /j,  in 
the  other  a  weight  q,  required  the  true  weight  of  the  body. 

Suppose  the  lengths  of  the  arms  of  the  balance  to  be  a  and  b,  and 
let  X  represent  the  true  weight  of  the  body,  then  its  moment  in  one 
scale  is  ax,  and  in  the  other  bx,  and,  by  the  problem, 

ax^bp,  bx=aq ; 
multiplying    these   together    we    have,   abx^  =  abpq  .'.  x^  y/pq; 
hence  the  true  weight  is  a  mean  proportional  between  the  two  false 
weights. 

Problem  II. — (78.)  The  common  steelyard  (fig.  63)  is  a  bar  AB, 
moveable  about  a  fulcrum  O  ;  P,  the  body  to  be  weighed,  is  hung 
at  the  shorter  arm  A,  and  a  given  weight  W  is  moved  along  the  other 
arm  till  it  balances  P  ;  then  the  weight  of  P  is  known  from  the  place 
of  W ;  to  find  how  the  distances  OC  increase  with  regard  to  the 
weight  P. 

Let  G  be  the  centre  of  gravity  of  AB,  and  Gg  vertical  meeting 
the  horizontal  line  AOC  in  g.  Call  the  weight  of  AB,  w,  OA.=p, 
OC=Pj,  Og=g,  then,  by  equation  (2)  above, 

IDcr  Pp 


ON  THE  MECHANICAL  POWERS.  95 

hence,  if  we  take  Ox  =-^y  we  have  3^0=-^^  ;  hence  xC  varies 
'  W  W 

as  P ;  and,  proceeding  from  x,  equal  additions  of  distance  to  xG 
correspond  to  equal  additions  of  weight  to  P :  that  is,  if  xQ>  be 
graduated  with  points  1,2,3,  &c.  at  equal  successive  intervals  from 
X,  W  placed  at  these  points  will  successively  balance  the  weights 
1,  2,  3,  &c. 

Problem  III. — (79.)  A  homogeneous  lever  AB  (fig.  64)  of  the 
second  kind  is  equally  thick  throughout;  it  is  required  to  deter- 
mine what  must  be  the  length  of  the  arm  AB,  that  a  given  weight 
W,  acting  at  the  extremity  of  the  arm  AC,  may  be  supported  by 
the  least  power  P  possible,  taking  into  account  the  weight  of  the 
lever  itself. 

The  lever  being  homogeneous,  and  equally  thick  throughout  any 
portion  of  its  length,  may  be  taken  to  represent  the  weight  of  that 
portion  ;  hence,  calling  the  length  AB  of  the  lever  x,  its  weight  will 
be  X,  acting  at  G,  its  middle  point;  therefore,  putting  AC=/;,  we 
have,  by  equation  (2), 

W  »,+ ^a;  •  x=Vx .-.  P=  ^-i-f  d  x. 
^  X 

To  determine  for  what  value  of  x  this  expression  for  P  is  a  mini- 

rf  P  W» 

mum,  we  have  —, — = f^  4-^=0 ;  from  which  we  immediate- 

dx  x^ 

ly  geta;^=2W/?j  .•.  x=y/\2  Vf  p^],  the  value  required. 

It  must  be  remembered  that  we  have  taken  lengths  to  represent 
weights,  so  that,  whatever  our  unit  of  length  has  been,  the  M^eight 
of  that  unit  must  be  considered  as  the  unit  of  weight,  thus  if  we  have 
measured/)^,  and  therefore  x,  in  inches,  then  the  numerical  expres- 
sion for  W  will  be  the  weight  W,  divided  by  the  weight  of  an  inch 
of  the  lever. 

By  substituting  this  value  of  x  in  the  foregoing  expression  for  P, 
we  have,  for  the  intensity  of  this  power  when  acting  at  the  greatest 
advantage, 

P=     :^p=+W{'i'^PA=^/\^'^P^]' 
V^2  W  j9j 

The  Wheel  and  Jlxle. 

(80.)  This  machine  is  in  reality  only  a  modification  of  the  lever; 
it  consists  of  two  parts,  a  cylinder  called  the  axle,  and  the  surround- 
ing circle  or  wheel  connected  with  it,  having  its  centre  in  the  axis 
of  the  cylinder  about  which  the  whole  turns  (see  fig.  65). 


96  ELEMENTS   OF   STATICS. 

This  machine  is  not  complete  in  itself  like  the  lever,  requiring  the 
addition  of  cords,  or  chains,  or  some  other  intermediate  body,  to 
communicate  with  the  forces  engaged. 

The  more  immediate  object  of  this  machine  is  to  support  or  raise 
a  weight  W,  suspended  to  a  rope  wound  about  the  axle,  by  means  of  a 
power  P  applied  to  the  circumference  of  the  wheel.  But  in  number- 
less applications  of  this  machine  the  axle  does  not  communicate  di- 
rectly with  the  weight  by  a  cord :  but,  by  being  surrounded  with  teeth, 
as  in  fig.  66,  acts  upon  a  toothed  wheel,  and  the  axle  of  this  last 
upon  another  toothed  wheel,  and  so  on,  the  power  being  thus  trans- 
mitted to  the  weight  W.  The  effect  of  a  power  thus  transmitted 
we  shall  consider  presently,  examining  first  the  more  simple  case 
just  noticed. 

Let  the  radius  of  the  wheel  be  p,  that  of  the  axle  p^,  then  (72) 

W      p 

1      Pi 

the  weight  and  power  being  to  each  other,  reciprocally,  as  their  dis- 
tances from  the  fixed  axis,  as  in  the  lever. 

If  the  equilibrating  forces  do  not  act  tangentially,  as  we  here  sup- 
pose, then  instead  o{ p,  p^  representing  the  radii,  they  will  represent 
the  distances  of  the  directions  of  the  power  and  weight  from  the 
axis. 

It  is  obvious  that  the  greater  be  the  wheel  the  less  will  be  the 
power  requisite  to  support  or  move  a  given  weight :  and  that  a  con- 
tinually decreasing  power  may  have  a  uniform  effect  upon  a  constant 
weight,  it  must  act  upon  a  series  of  continually  increasing  wheels, 
constantly  keeping  up  the  above  proportion,  the  radii  of  the  wheels 
varying  inversely  as  the  powers:  thus  P  and  P'  being  any  two 
%'alues  of  the  powers  we  have 

—  -t.   ^-  Pi       E.  _Z' 

^  ~  Pi'¥i~  Px"  P'~^' 

In  this  way  the  varying  power  exerted  by  the  main-spring  of  a 

watch  while  uncoiling  is  made  to  produce  a  uniform  efl^ect ;  this 

power  acting  on  a  series  of  varying  wheels  (fig.  67)  called  ihefuzee. 

It  should  be  remarked  that  the  macliine  we  are  now  considering 
is  virtually  unchanged,  though  the  wheel  be  stripped  of  its  rim  and 
the  power  be  applied  at  the  extremities  of  the  spokes. 

It  should  be  further  remarked  that  the  thickness  of  the  rope,  when 
considerable,  must  not  be  neglected  in  estimating  the  conditions  of 
equilibrium ;  for  we  ought  to  consider  the  forces  to  be  transmitted 
along  the  middle  or  axis  of  the  rope,  and,  therefore,  the  radius  of 
the  rope  should  be  added  to  that  of  the  wheel,  and  of  the  axle,  so 
that  in  the  above  expressions  p  and  p^  are  the  distances  of  the  axes 
of  the  ropes  from  the  axis  of  the  machine. 


ON    THE    MECHANICAL    POWERS.  97 

(81.)  When  the  axle  is  toothed  it  is  called  a  ;jin«on,  and  the  teeth 
its  leaves.  If  a  power  be  in  equilibrium  with  a  weight  by  means 
of  a  system  of  toothed  wheels  and  pinions,  as  in  fig.  66,  then  we 
shall  iind  that  the  power  will  be  to  the  weight  as  the  product  of  the 
radii  of  the  pinions  to  the  product  of  the  radii  of  the  wheels. 

For  let  the  radii  of  the  axle  or  pinions  be  r,  r^,  r^,  &c.  and  those 
of  the  wheels  R,  R^,  R^,  &c.  then  the  power  P  acting  on  the  first 
wheel  equilibrates  a  weight  or  power  P^,  on  the  pinion,  expressed 

PR 

by  Pj= ;  this  then  is  the  power  applied  to  the  second  wheel 

and  which,  therefore,  comnaunicates  to  its  pinion  a  power  expressed 

by  P.=  -^^  =  ZM.; 

r^  r  r^ 

this   IS  ttie  power  applied  to  the  third  wheel,  and  continuing  this 
computation  it  is  plain  that  the  power  which  the  nth  pinion  or  axle 

Tk      P  R  R,  R„  ....  Rn_i         ,  1     T>,     •     1 

acts  IS  P„  =  - — ' ;  and  consequently,  Vn.  is  the 

r  r^T^  .  .  .  .  Tn—\ 

weight  which  the  power  P  will  balance  on  the  wth  axle. 

Instead  of  expressing  this  result  in  words,  as  above,  we  may  say, 
since  the  radii  are  as  the  circumference,  and  these,  again,  as  the 
number  of  teeth  they  carry,  that  the  power  is  to  the  wciight  as  the 
product  of  the  numbers  expressing  the  leaves  to  each  pinion  to  the 
product  of  the  numbers  expressing  the  teeth  to  each  wheel ;  the 
number  for  the  first  wheel,  which  is  plane,  expressing  the  number 
of  teeth  it  could  carry,  and,  in  like  manner,  the  number  for  the  last 
axle  being  that  expressing  the  number  of  leaves  it  would  carry. 

For  further  particulars  respecting  tooth  and  pinion  work,  and,  in- 
deed, respecting  machinery  in  general,  the  student  is  referred  to 
Professor  Gregory's  Treatise  of  Mechanics,  a  work  abounding  with 
valuable  information.  Much  interesting  matter  will  also  be  found 
in  Dr.  Lardner's  elegant  volume  on  Mechanics,  in  the  Cabinet  Cy- 
clopaedia. 


SCHOLIUM. 

(82.)  The  student  will  have  observed  that  the  foregoing  theory 
of  toothed  wheels  is  founded  on  the  supposition  that  the  power 
communicated  from  the  tooth  of  one  wheel  to  that  of  another  is  in 
the  direction  of  a  tangent  to  the  circle  on  which  this  latter  is  raised, 
as  in  the  plane  wheel  and  axle,  and  that  such  may  really  be  the  di- 
rection of  the  power,  a  particular  figure  must  be  given  to  the  teeth, 
at  least  to  the  working  sides  of  them.  This  figure  is  that  of  the  invo- 
lute of  the  circle  on  which  they  are  raised.  Thus,  let  IHF,  KE6 
I  13 


98  ELEMENTS   OF    STATICS. 

(fig.  68)  be  the  wheels  to  whicli  the  teeth  are  to  be  accommodated, 
the  acting  face  GCH  of  the  tooth  a  must  have  the  form  of  tlie  curve 
traced  by  the  extremity  II  of  the  flexible  line  F«H,  as  it  is  unwrap- 
ped from  the  circumference  ;  and,  in  like  manner,  the  acting  face  of 
the  tooth  b  must  be  formed  by  the  unwrapping  of  a  thread  i'rom  the 
circumference  of  the  circle  KE6.  T!ie  line  FCE  drawn  to  touch 
both  circles  will  cut  the  surfaces  of  the  two  teeth  in  C,  the  point 
where  they  touch  each  other,  at  a  point  in  the  common  tangent  to 
both  circles,  and  the  force  arising  from  their  mutual  pressure  will 
always  act  in  the  direction  of  the  circumference  of  the  wheels  at  E 
and  F.  But,  continues  Dr.  Gregory,  whose  words  we  have  bor- 
rowed in  the  preceding  description,  although  Roemer,  Varignon,  De 
la  Hire,  Camus,  Euler,  Emerson,  Kaestner,  and  Robison,  have 
turned  their  thoughts  to  this  object,  and  some  of  them  have  stmck 
out  rules  of  ready  application  in  practice,  it  is  to  be  regretted  that 
these  rules  have  been  little  followed  by  practical  mechanics,  most  of 
whom  have,  in  this  case,  been  more  inclined  to  follow  a  set  of  hack- 
iiied  rules  handed  down  from  one  workman  to  another,  although  com- 
pletely destitute  of  scientific  principle.  Even  watchmakers,  in  whose 
constructions  a  little  more  than  common  skill  and  nicety  in  the  exe- 
cution might  be  expected,  are  but  few  of  them  acquainted  with  any 
rules  founded  upon  the  deductions  of  accurate  theory ;  but  commonly, 
we  are  informed,  give  to  their  teeth  the  shape  assumed  by  a  horse 
hair  when  held  bent  between  the  fingers,  a  method  so  vague  that  it 
is  difficult  to  conceive  how  it  came  to  be  adopted. 

The  Pulley. 

(83.)  A  pulley  is  a  grooved  wheel  moving  freely  on  an  axis,  and 
fixed  in  a  case  or  block.  It  communicates  applied  force  in  conjunc- 
tion with  a  cord  which  the  groove  receives. 

The  Jixed  pulley  we  have  already  employed  in  various  parts  of 
this  work  for  the  sole  purpose  to  which  it  is  applicable,  viz.  for  the 
purpose  of  changing  the  direction  of  a  force  acting  by  a  cord,  and 
although  the  fixed  pulley  is,  for  this  purpose  even,  an  important 
instrument,  yet  as  it  docs  not  afford  any  mechanical  advantage  in 
the  way  of  accumulating  force  it  offers  no  theory  for  discussion.  It 
is  different  with  the  moveable  pulley  (fig.  60,)  to  the  block  of  which 
the  weight  is  fastened,  which  is  sustained  between  the  power  P, 
acting  at  one  end  of  the  cord,  and  the  pressure  on  a  fixed  hook  Q  at 
the  other  end ;  the  tension  of  the  cord  being  uniform,  it  is  obvious 
that  the  intensity  of  P  must  be  equal  to  the  strain  or  pressure  on  Q. 
Let  us  examine  the  relation  which  these  equal  powers  bear  to  the 
weight.  Continue  the  directions  of  the  ropes  PC,  QD,  till  they 
meet,  unless  they  should  be  parallel ;   then,  since  the  system  of 


ON    THE    MECHANICAL    POWERS.  99 

forces  P,  Q,  W  is  in  equilibrium,  the  point  E,  wliere  the  directions 
of  two  meet,  must  be  in  the  direction  of  the  third,  and  the  resultant 
W  of  the  equal  forces  P,  Q  will  be 

W=2P  cos.  a  ....  (1); 
a  being  equal  to  half  the  inclination  of  the  cords  from  P  and  Q. 
Instead  of  the  tabular  cosine  we  may  introduce  into  this  expression 
the  cosine  corresponding  to  the  radius  r  of  the  pulley,  provided  we 

write  '- —  instead  of  cos.  a ;  this  cosine  is,  obviously,  the  line 

r 

C??,  being  the  sine  of  the  angle  COn  the  complement  of  a;  hence, 

dividing  each  side  of  the  expression  thus  changed  by  P,  we  have 

W     2  cos.  a 


P  r        •      •  •  v-/' 

that  is,  the  power  is  to  the  weight  as  the  radius  of  the  pulley  to  the 
chord  of  that  arc  of  it  which  is  in  contact  with  the  rope. 

If  the  cords  from  P  and  Q  are  parallel,  then  the  equilibrium  being 
the  effect  of  three  parallel  forces,  the  middle  force  W  must  be  equal 
to  the  two  P  and  Q;  in  this  case,  therefore,  W=2P  .  (3);  and  it 
i.3  obvious  that  half  the  circumference  of  the  pulley  must  be  in  con- 
tact with  the  rope. 

It  may  be  observed  from  the  expression  (1)  that  tiie  moveable 
pulley  affords  a  mechanical  advantage  only  so  long  as  the  inclina- 
tion of  the  cords  from  P  and  Q  is  less  than  120°;  for  at  this  angle 
cos.  a=5,  and  at  greater  angles  cos.  a  is  less  than  5.  The  great- 
est advantage  is  gained  when  the  cords  are  parallel. 

(84.)  The  advantage  gained  by  a  single  moveable  pulley  may  be 
multiplied  to  any  extent  by  employing  a  system  of  pullies,  as  in 
fig.  70,  thus,  representing  the  tensions  of  the  several  ropes  by  the  let- 
ters annexed  to  them  in  the  figure,  we  have  from  equation  (1)  above 
\Y=2t,  cos."^a 
t^  =2  t^  cos.  ftg 
ta  =2  t^  cos.  a. 


tn-i=^t„  cos.  a„=2  P  cos.  a„  ; 
hence,  multiplying  these   equations   together  and  expunging  the 
factors  common  to  each  side  of  the  resulting  equation,  we  have 
W=2"  P  (cos.  a.  cos.  a^.  COS.  a^  .  .  .  .  COS.  a„)    ....    (4). 
P  being  the  power  and  n  the  number  of  pullies. 

If  the  several  cords  are  parallel,  as  in  fig.  71,  this  equation  be- 
comes W=2"  P  (5) ;  and,  if  the  angles  are  all  equal  to  each  other, 
W=2«P  cos.^tt  ....  (6). 

In  what  is  here  said  the  weights  of  the  several  pullies  have  been 


100  ELEMENTS   OF    STATICS. 

neglected,  but  in  strictness  these  should  be  taken  into  account.  If 
they  increase  the  weights  on  the  ropes  which  pass  round  them  by 
the  several  quantities  A,  A,,  A^,  «fcc.,  then  the  above  series  of  equa- 
tions will  be 

W  +  A  =2/,  cos.  o, 

/,  -f  Aj=2/^cos.  a^ 

&c.  &c. 

(85.)  Instead  of  attaching  the  several  ropes  to  immoveable  points, 
as  in  fig.  71,  they  are  all  in  another  arrangement  of  the  system 
fastened  to  the  weight,  as  in  fig.  72,  the  ropes  being  parallel.  The 
relation  between  the  power  and  weiglit  in  this  system  is  at  once 
seen  from  looking  at  the  figure ;  thus  the  first  pulley  or  that  which 
first  receives  the  power  supports  twice  P,  the  second,  therefore, 
supports  four  times  P,  the  third  eight  times  P,  and  the  nth  supports 
2"  times  P,  and  all  of  this  except  P  is  tlie  weight ;  hence  deducting 
P  the  weight  supported  is  W  =  (2«  —  1)  P.. .  .  (7). 

In  this  system,  as  well  as  in  tliat  exhibited  in  fig.  71,  each  pul- 
ley is  connected  with  two  parallel  branches  of  rope,  of  which  each 
branch  bears  half  the  weight  attached  to  the  pulley  ;  but  if  three 
parallel  branches  of  rope  be  connected  with  each  pulley,  as  in  the 
systems  exhibited  in  figures  73  and  74,  each  branch  will  bear  a 
third  of  the  attached  weight ;  hence,  if  ?i  be  the  number  of  these 
systems  of  threes,  we  shall  have  instead  of  the  equations  (5)  and  (7), 

W=3''  P  and  W  =  (3''  — 1)....P; 
the  first  equation  belonging  to  the  arrangement  in  fig.  73,  and  the 
second  to  that  in  fig.  74. 

(83.)  We  have  hitherto  supposed  each  pulley  to  be  attached  to  a 
separate  string ;  but  if  only  one  string  is  employed,  as  in  the  sys- 
tems represented  in  fig.  75,  then,  as  this  string  is  uniformly  tense 
throughout,  the  common  tension  being  P,  the  weight,  which  is 
equal  to  the  sum  of  the  tensions  when  the  branches  of  the  rope  are 
parallel,  must  be  equal  to  2P  times  the  number  of  moveable  puUies ; 
that  is,  W=2n  P,  W  including  the  weight  of  the  lower  block. 

77ie  Inclined  Plane. 

(87.)  This  machine  is  simply  a  plane  surface  inclined  to  the  ho- 
rizon. 

The  theory  of  this  machine  is  unfolded  in  problem  I.  page  48, 

Avhere  it  is  shown  that  if  i  be  the  inclination  of  the  plane,  and  e  the 

angle  which  the  direction  of  the  power  makes  with  the  plane,  the 

sin.  t 
relation  between  the  power  and  weight  is  P=W '■ — .       When 


ON    THE    MECHANICAL    POWERS,  101 

the  direction  of  the  power  is  parallel  to  the  plane,  the  machine  af- 
fords the  greatest  mechanical  advantage  possible,  for  then 

COS.  £  =  1  and  P=W  sin.  i, 
so  that  in  this  case  the  power  is  to  the  weight  as  the  altitude  of  the 
plane  to  its  length. 

But  when  the  direction  of  the  power  is  horizontal  or  parallel  to 
the  base  of  the  plane  then  £=i,  and  the  relation  is  P=W  tan.  z, 
which  shows  that  the  power  is  to  the  weight  as  the  height  of  the 
plane  to  the  base. 

The  Screiv. 

(88.)  Before  we  can  investigate  the  mechanical  power  of  the 
screw  we  must  ascertain  exactly  the  form  of  the  spiral  surface 
which  it  presents.  The  generation  and  equation  of  this  surface  has 
been  explained  in  the  Diff".  Calc.  p.  200  ;  but  it  Avill  be  proper  to 
repeat,  in  part,  that  explanation  in  this  place.  Let  us  conceive  then 
a  rectangle  to  be  rolled  round  a  vertical  and  cylindrical  column 
which  it  just  embraces,  the  line  which  was  the  diagonal  of  this 
rectangle  will  form  itself  into  a  winding  curve,  called  a  helix,  and 
it  will  make  just  one  turn  round  the  column,  its  horizontal  projec- 
tion being  a  circle  ;  if  immediately  above  this  another  equal  rectan- 
gle be  applied  to  the  cylinder,  the  vertical  edges,  when  brought 
together,  being  in  a  line  with  those  of  the  first,  the  diagonal  of  this 
will  form  a  continuation  of  the  helix,  and,  in  this  way,  will  be  ex- 
hibited on  the  surface  of  the  cylinder,  the  trace  of  the  winding  sur- 
face which  forms  the  screw. 

If  now,  beginning  with  the  bottom,  we  were  to  strip  off  these 

rectangles  one  after  the  other,  turning  the  cylinder  round  at  the 

same  time,  so  that  all  of  tliem  might  be  ranged  in  the  same  vertical 

plane,  we  should,  obviously,  have  the  figure  presented  at  fig.  76, 

tlie  uniform  straight  line  AB  being  the  developement  of  the  helix ; 

we  may,  therefore,  say  that  this  curve  is  formed  by  winding  round 

an  upright  cylinder  an  inclined  straight  line  AB,  always  preserving 

its  inclination  constant ;  if  we  consider  this  inclined  line  to  be  the 

edge  of  an  inclined  plane,  then  the  surface,  thus  wound  round  the 

cylinder,  will  be  the  surface  of  the  screw.     This  surface,  therefore, 

differs  from  the  inclined  plane  in  no  respect  but  in  its  winding 

course,  and  it  will,  obviously,  require  just  as  much  power  to  sustain 

a  weight  on  the  winding  surface  as  on  the  straight  surface,  so  that 

if  W  be  any  weight  on  the  surface,  and  a  power  acting  horizontally 

P     BC       be       ^       *     .      , 
support  it,  we  must  have  —-  =  ——=— — ;    but  Ac  is  the  circum- 
W     AC      Ac 

ference  of  the  cylinder  which  carries  the  screw,  therefore  calling 

the  radius  of  it  r,  and  the  height  be,  which  measures  the  interval 

between  each  turn  of  the  screw  and  the  next,  h,  we  have 

i2 


102  ELEMENTS    OF    STATICS. 

but  if  the  power  P,  instead  of  being  applied  directly  to  W,  is  ap- 
plied to  the  arm  of  a  lever,  at  R  distance  from  the  fulcmm,  and  if 
the  direction  of  W  be  at  r  distance  on  the  other  side,  then  the  value 

of  P  Avill  be  P — ,  and  in  this  case  the  condition  of  equilibrium  will 
r 

be  P-=W  JL-  ...  P=W  J^  ....  (2). 

r  2  7t  r  2  rt  K 


Now  this  equation  expresses  the  power  of  tlie  screw ;  for  the 
weight  to  be  balanced  or  raised,  the  resistance  to  be  overcome,  the 
pressure  to  be  sustained,  &c.  is  always  a  force  in  the  direction  of 
the  axis  of  the  cylinder,  and  acting  upon  the  inclined  plane  wind- 
ing round  it;  it  is  balanced  by  a  power  P,  acting  perpendicular  to 
the  same  axis,  at  the  extremity  of  a  lever  (see  tig.  77)  whose  ful- 
crum is  in  the  axis,  and,  therefore,  at  the  distance  of  the  radius  of 
the  cylinder  from  the  winding  surface  or  thread  of  the  screw.  The 
weight  is  throAvn  on  the  thread  by  its  being  connected  with  a  nut 
or  internal  screw  N,  which  is  a  spiral-grooved  case  fitted  to  receive 
the  external  screw. 

In  fig.  77,  the  whole  pressure  on  the  screw  is  thrown  upon  the 
thread  within  the  nut  N,  and  the  power  applied  at  the  extremity 
of  the  lever  R  must  bear  to  this  pressure  or  weight  the  relation  (2) 
above,  in  order  to  balance  it,  or,  which  is  the  same  thing,  a  power 
something  greater  than  this  must  be  applied  to  move  the  lever. 
The  relation  (2)  when  expressed  in  words  is  this,  viz.  The  power 
is  to  the  \veight  or  resistance  as  the  interval  between  two  adjacent 
turns  of  the  thread  to  the  circumference  of  the  circle  described  by 
the  power ;  and  this  relation  we  see  is  altogether  independent  of 
the  radius  of  the  cylinder,  and,  therefore,  also  of  the  degree  of  pro- 
tuberance of  the  thread ;  it  varies  only  with  the  distance  h  between 
the  turns  or  contiguous  spires  of  the  thread,  and  with  the  inclina- 
tion of  the  thread  to  the  axis  of  the  cylinder ;  hence  so  long  as  this 
distance  and  this  inclination  is  preserved,  it  matters  not  what  form 
be  given  to  the  surface  of  the  thread,  nor  how  protuberant  it  be 
made. 

We  see  from  the  expression  (2)  that  there  are  two  ways  in  which 
the  power  of  the  screw  may  be  increased ;  first  by  diminishing  the 
distance  h  between  the  turns,  or,  secondly,  by  increasing  the  length 
of  the  lever  R ;  it  is,  however,  not  strictly  correct  to  say  that  the 
power  of  the  screw  is  increased  by  this  latter  change,  for  this  in 
fact  remains  unaltered :  it  is  the  applied  power  that  is  here  in- 
creased. 


ON  THE  MECHANICAL  POWERS.  103 

The  Wedge. 

(89.)  The  wedge  is  a  triangular  prism,  AC  (fig.  78),  chiefly  em- 
ployed for  splitting  or  separating  bodies  ;  for  this  purpose  the  edge 
AB  of  the  wedge  is  introduced  between  the  bodies  to  be  separated, 
and  the  power  which  drives  it  is  applied  to  the  head  DC  ;  this 
power,  when  the  machine  is  in  equilibrium,  must  balance  the  re- 
sistances opposed  to  its  entry,  and  which  can  only  act  on  the  edge 
AB,  and  on  the  two  faces  DB,  CA.  Indeed  in  a  state  of  equili- 
brium there  can  be  no  resistance  opposed  to  the  edge  of  the  instru- 
ment, as  is  obvious,  so  that  this  state  is  preserved  by  three  forces, 
viz.  the  power  P  applied  to  the  head,  and  the  resistances  P^,  P^ 
acting  against  the  faces.  To  determine  the  conditions  of  equili- 
brium of  forces,  thus  acting,  we  must  introduce  hypotheses,  not 
only  unsupported  by  observation  and  experiment,  but  in  direct  con- 
tradiction of  them ;  thus,  for  three  forces  to  keep  a  body  in  equili- 
brium, it  is  absolutely  necessary  that  when  one  of  the  three  is 
withdrawn,  the  body,  through  the  influence  of  the  other  two,  should 
move  unmolested  in  a  direction  opposite  to  that  of  the  force  with- 
drawn ;  in  the  wedge,  therefore,  when  the  pressure  or  impelling 
power  is  withdrawn  from  the  head  the  pressures  on  the  sides  should 
expel  it  from  between  the  resisting  surfaces ;  this,  howevei',  is  al- 
most universally  contrary  to  experience  :  in  the  cleaving  of  wood, 
for  instance,  the  wedge  may  be  driven  to  any  extent  between  the 
resisting  sides  (fig.  79),  and  will  usually  remain  there  without  sen- 
sibly receding,  although  the  power  be  removed  from  the  head,  the 
friction  being  fully  equal  to  balance  the  expelling  forces  acting  on 
the  faces. 

The  mathematical  theory  of  this  machine  is  founded  on  the 
hypothesis,  that  the  resisting  surfaces  as  well  as  the  faces  of  the 
wedge  are  perfectly  smooth,  or,  which  is  the  same  thing,  that  the 
friction  is  nothing,  whereas  in  practice  the  friction  is  every  thing, 
the  wedge  would  be  comparatively  useless  without  it.  Seeing, 
therefore,  the  great  eflfect  of  friction  in  this  machine,  which  is  suf- 
ficient to  maintain  the  equilibrium  even  Avhen  the  applied  power  is 
withdrawn,  it  is  obvious  that  no  deduction  from  the  mathematical 
theory  of  it  can  be  of  much  practical  utility.  It  is  true,  indeed, 
that  in  all  other  machines,  as  well  as  in  the  wedge,  friction  always 
opposes  a  hindrance  to  the  full  effect  of  the  applied  powers  to  pro- 
duce motion,  and  that,  therefore,  the  deductions  of  pure  theory, 
where  these  hindrances  are  not  taken  into  account,  require  some 
modification  before  they  can  agree  with  the  results  of  actual  expe- 
riment, but  then,  by  polishing  or  lubricating  the  acting  surfaces, 
these  hindrances  may  be  more  and  more  diminished,  and  the  results 
of  practice  be  made  to  approa^'ti  nearer  and  nearer  to  the  deductions 


104  ELEMENTS  OF  STATICS. 

of  theory.  In  the  application  of  the  wedge,  however,  it  is  in  most 
cases  impossible,  even  if  it  were  desirable  to  diminish  in  the  small- 
est degree  the  friction  on  its  faces,  it  is  this  that  hinders  the  pres- 
sures on  the  faces  from  driving  the  wedge  back,  and  is,  therefore, 
a  power  which  greatly  favours  the  efficacy  of  the  machine ;  as  the 
impelling  force  is  usually  applied  at  intervals,  by  means  of  repeated 
blows,  and  not  in  the  form  of  a  continued  pressure,  the  friction 
serves  to  hold  the  wedge  where  the  last  blow  had  driven  it. 

(90.)  Abstracting  from  the  influence  of  friction,  the  equilibrium 
of  the  wedge  may  be  thus  investigated.  Let  any  arbitrary  length, 
DE,  (fig.  80,)  represent  the  power  applied  perpendicularly  to  the 
head  of  the  wedge,  and  draw  DM,  DN  perpendicular  to  the  faces 
AC,  BC,  and  complete  the  parallelogram  IK,  having  DE  for  its 
diagonal ;  DI  and  DK,  or  IE,  will  then  represent  the  pressures  P^, 
Pj  against  the  faces  of  the  wedge,  so  that  the  three  equilibrating 
powers  are  as  the  three  sides  of  the  triangle  DEI,  or  as  the  three 
sides  of  the  similar  triangle  ABC,  that  is, 

P  :  P,  :  P,  :  :  AB  :  AC  :  BC  ; 
or,  calling  the  length  of  the  edge  C,  /, 

P  :  P,  :  P,  :  :  ABx/ :  ACx/:  BCx/; 
which  proportion,  obviously,  implies  that  the  power  on  the  head 
of  the  wedge  and  the  equilibrating  pressures  on  the  faces  are  pro- 
portional to  the  areas  of  the  head  and  faces,  on  which  they  re- 
spectively act. 

SCHOLIUM. 

To  the  foregoing  theory  of  the  simple  machines  we  shall  append 
the  following  judicious  remarks  from  Venturoli. 

The  false  opinion  which  persons  unskilled  in  the  nature  and  the 
power  of  machines  are  apt  to  conceive,  often  encourages  empty 
errors  and  mischievous  deceptions.  One  of  the  most  common  of 
these  conceits  is  that  of  considering  machines  as  available  to  in- 
crease and  multiply  the  force  of  agents,  which  is  not  always  true. 
To  form  a  just  notion  of  the  aid  which  may  be  expected  from  ma- 
chines, looking  to  the  uses  to  which  they  are  most  commonly  put, 
we  shall  divide  them  into  two  classes ;  those  intended  simply  to 
sustain  a  weight,  and  those  intended  to  draw  it,  or  raise  it  equably. 

In  machines  of  the  first  class,  both  the  effect  of  the  machine  and 
the  immediate  effect  of  the  power  can  only  be  estimated  by  the 
weight  sustained.  This  being  understood  it  is  evident  that  the 
machine  increases  the  eflfect  of  the  power ;  so  that,  for  example, 
a  force  of  10  lbs.  will  sustain  by  means  of  a  lever,  100  lbs.,  pro- 
vided that  the  arm  of  the  force  be  ten  times  as  long  as  that  of  the 
weight. 


ox  THE   MECHANICAL  POWERS.  105 

If  it  be  asked  how  the  force  can  ever  produce  an  effect  so  much 
greater  than  itself,  we  shall  perceive,  if  we  consider  well,  that  the 
force  10  does  not  really  sustain  the  whole  weight  100,  but  only  the 
tenth  part  of  it.  Let  the  lever  be  supposed  to  be  of  the  second  kind  ; 
the  force  100  may  be  resolved  into  two,  the  one  equal  to  90  which 
acts  upon  the  fulcrum,  and  the  other  equal  to  10  which  acts  at  the 
point  of  application  of  the  power.  The  first  is  entirely  sustained 
by  the  prop,  and  the  power  sustains  the  second  alone.  Archimedes 
required  only  a  fixed  point  to  hold  the  terraqueous  globe  in  equili- 
brium. If  he  had  found  it,  says  Carnot,  it  would  not  in  reality  have 
been  Archimedes,  but  the  fixed  point,  Avhich  would  have  sustained 
the  earth. 

In  machines  of  the  second  class  neither  the  effects  of  the  machine 
nor  that  of  the  power  can  be  estimated  simply  by  the  weight  raised  ; 
otherwise  the  measure  of  the  effect  would  be  altogether  vague  and 
indeterminate.  In  fact  any  force,  however  small,  may  carry  a  weight 
of  any  assignable  magnitude  however  great ;  if  it  only  be  granted 
that  the  weight  admits  of  being  divided  and  of  being  carried,  one 
piece  at  a  time.  Wherefore  it  is  necessary  to  take  into  account 
the  time  also  in  which  the  power  can  carry  the  weight  through  a 
given  space,  or  the  velocity  with  which  the  weight  is  carried  ;  and  on 
this  account  it  is  that  the  effect  is  measured  by  the  product  of  the 
weight  multiplied  by  the  velocity. 

Now  upon  this  principle  we  have  already  shown  that  the  machine 
does  not  increase  the  effect  of  the  force.  If  a  man  with  a  force  equi- 
valent to  10,  raise,  by  means  of  a  machine,  a  weight  of  100,  he 
moves  with  a  velocity  ten  times  as  great  as  that  of  the  weight,  and 
does  as  much  as  if  operating  without  any  machine  he  carried  those 
100  at  ten  journeys,  loading  himself  with  10  at  a  time.  In  a  word, 
what  is  gained  in  the  quantity  of  the  weight  moved  is  lost  in  the  ve- 
locity ;  and  the  effect  remains  the  same. 

Between  the  two  classes  of  machines,  above  described,  there  is  then 
this  characteristic  difference,  that  the  first  add  to  the  effect  of  the 
power,  the  second  do  not  add  to  it. 

There  is  another  difference,  not  less  remarkable,  respecting  the 
resistances  of  friction,  and  of  ropes,  and  other  resistances.  In  ma- 
chines of  the  first  class  these  resistances  are  all  of  them  advanta- 
geous to  the  power*  and  themselves  also  sustain  their  portion  of  the 
weight ;  whence  there  remains  so  much  the  less  of  it  for  the  power 
to  support.  On  the  contrary,  in  machines  of  the  second  class,  the 
resistances  are  all  of  them  detrimental  to  the  power,  and  form  part  of 
the  weight  to  be  overcome :   whence,  on  this  account,  a  force  is  re- 

*  Because  the  weight,  before  it  can  move,  must  overcome  these  resistances  as 
well  as  the  power. 

14 


106  ELEMENTS    OF    STATICS. 

quired  greater  than  that  which  would  be  required  in  the  immediate 
application  of  the  power. —  VenturoWs  Mechanics,  part  ii.  p.  164. 


CHAPTER  V. 

ON  THE  STRENGTH  AND  STRESS  OF  BEAMS. 

(91.)  It  is  obvious  that  in  all  the  practical  operations  of  me- 
chanics, and  more  especially  in  the  raising  of  structures,  it  is  of 
great  importance  to  know  the  weight  or  stress  each  component  part 
is  fitted  to  bear  without  endangering  the  stability  of  the  whole  ;  and, 
consequently,  numerous  experiments  to  ascertain  the  strength  of 
materials,  particularly  of  beams  and  bars,  have  at  various  times  been 
undertaken  by  scientific  men.  Into  a  detail  of  these  experiments  we 
do  not,  however,  propose  to  enter,  but  merely  to  present  to  the  student, 
in  a  short  compass,  some  of  the  more  interesting  and  valuable  par- 
ticulars furnished  by  theory,  and  confirmed  by  the  experiments  ad- 
verted to. 

If  a  uniform  rod  or  bar  of  any  substance  be  suspended  by  one  ex- 
tremity, and  loaded  at  the  other  till  it  is  on  the  point  of  being  torn 
asunder,  we  ought  to  expect,  independently  of  actual  experiment, 
that  the  weight  would  be  proportional  to  the  tranverse  section ;  for, 
if  this  bar  were  conceived  to  be  divided  longitudinally  into  any  num- 
ber of  equal  strips,  no  reason  could  be  assigned  why  one  of  these 
should  support  a  greater  portion  of  the  weight  than  either  of  the 
others,  so  that  each  would  support  an  equal  part  of  the  weight  sup- 
ported by  the  whole,  just  as  an  assemblage  of  parallel  ropes  divide 
the  weight  of  an  appended  body  equally  among  them.  AH  experi- 
ments on  lateral  strains  prove  this  deduction  to  be  correct,  and  to 
be  quite  independent  of  the  figure  of  the  section,  requiring  only 
uniformity  and  equality  in  the  texture  of  the  bodies  compared,  so 
that  we  may  lay  it  down,  as  a  general  law,  that  in  bars  of  the  same 
material  the  lateral  resistances  are  as  the  areas  of  their  traverse  sec- 
tions. 

(92.)  When  the  bar  or  beam  is  supported  in  a  horizontal  position, 
then  the  law  of  resistance,  which  it  opposes  to  fracture  by  an  in- 
cumbent weight,  is  more  difficult  to  establish,  because  here  we  do 
not  see  so  clearly  how  the  resisting  forces  exert  themselves,  nor  in 
what  degree.  It  was  laid  down  by  Galileo  that  if  a  beam  were  sup- 
ported at  its  extremities,  as  in  fig.  81,  and  loaded  by  a  weight  at  the 
middle,  that  all  the  fibres  of  the  beam  would  exert  equal  resistances 
to  prevent  fracture,  and  that  when  these  were  overcome  the  section 


THE  STRENGTH  AND  STRESS  OF  BEAMS.  107 

would  tend  to  turn  about  that  boundary  of  it  in  contact  with  the 
weight,  viz.  about  AB.  As  all  the  fibres  exert  equal  resistances, 
and  in  the  direction  of  their  lengths  these  resistances  will  be  so  many 
equal  and  parallel  forces  which  may,  therefore,  be  considered  as 
concentrated  in  the  centre  of  gravity  of  the  section,  so  that  denoting 
the  resistance  of  a  single  fibre  by  k,  and  considering  the  section  to 
be  a  rectangle  of  breadth  b,  and  depth  h,  kbh  will  express  the  sum 
of  the  resisting  forces,  and  as  this  acts  at  the  centre  of  gravity  its 
moment  to  turn  the  section  about  AB  will  be 

kbhx\h= — 

(93.)  The  hypothesis  of  Leibnitz  agreed  with  that  of  Galileo, 
as  regards  the  axis  about  which  the  section  would  turn,  but  it  dif- 
fered from  it  as  regards  the  equal  resistances  of  the  fibres  through- 
out the  whole  fracture ;  for,  according  to  Leibnitz,  the  forces  ex- 
erted by  the  fibres  were  directly  proportional  to  their  distances  from 
the  axis  of  the  section,  so  that  the  middle  fibre  exerted  but  half  the 
force  of  the  extreme  fibre,  therefore,  calling  the  force  of  this  k,  the 

sum  of  the  forces  would  be  ~-^,  and  the  centre  of  such  a  system 

of  parallel  forces  being  at  |/i,  the  moment  to  turn  the  section  would 
kbh^ 

~^: 

Now  it  may  be  remarked  that  as  far  as  regards  the  comparative 
strength  of  rectangular  beams  of  the  same  material,  or  of  beams 
generally,  which  have  only  rectangular  sections  when  cut  traversely 
it  matters  not  which  of  these  hypotheses  be  adopted,  for  both  equally 
warrant  the  inference  that  the  laiv  of  resistance  is  as  the  breadth, 
multiplied  by  the  square  of  the  height  or  depth;  and  this  law, 
which  has  been  confirmed  by  numerous  experiments,  immediately 
leads  to  an  inference  of  considerable  practical  importance,  viz.  that 
a  beam  is  much  more  efficient  when  placed  with  its  narrower  side 
uppermost,  that  is,  so  that  its  breadth  may  be  less  than  its  depth  ; 
for  if  we  call  the  breadth  b,  and  the  depth  or  height  h,  then  the  re- 
lative strengths,  when  b  and  h  are  alternately  uppermost,  are  ex- 
pressed by  bh^  and  hb^,  and,  consequently,  in  the  former  position 

the  beam  is  —  times  as  strong  as  in  the  latter,  that  is,  as  many  times 

as  strong  as  the  depth  contains  the  breadth.  This  important  fact  is 
always  attended  to  in  buildings,  the  joist,  rafters,  &c.  being  always 
placed  with  the  narrower  side  uppermost. 

It  has  been  supposed  above  that  the  segments  of  a  fractured  beam 
tend  to  turn  about  the  line  where  the  fracture  terminates  ;  but,  from 


108  ELEMENTS    OF    STATICS. 

f'xperimpnts  recently  undertaken  by  Mr.  Barlow,  it  appears  that  AB 
(fig.  82)  is  not  the  line  about  which  the  section  tends  to  turn,  as  Ga- 
lileo and  Leibnitz  had  supposed,  but  that  the  tendency  is  to  turn 
about  a  line  entirely  within  the  section,  so  that  the  fibres  on  that 
side  of  the  line  where  the  fracture  begins  are  extended,  and  those  on 
the  other  side  compressed ;  this  axis  Mr.  Barlow  calls  the  ncnlrcd 
axis,  dividing  the  section  into  the  area  of  tension  and  the  area  of 
compression,  and  he  calls  the  centre  of  tension  or  of  compression 
that  point  in  the  area  of  tqnsion  or  of  compression  where  all  the 
forces  in  that  area  should  be  collected  to  have  the  same  effect,  or  the 
same  moment,  with  respect  to  the  neutral  axis. 

The  existence  of  a  neutral  axis  somewhere  within  the  area  of  frac- 
ture, was  maintained  by  Mariotte,  James  Bernoulli,  and  Professor 
Robison ;  but  Mr.  Barlow  appears  to  have  been  the  first  who  set 
about  the  determination  of  this  axis  by  actual  experiment.  (See  the 
historical  sketch  of  former  theories,  prefixed  to  Mr.  Barlow's  Essay 
the  Strength  and  Stress  of  Timber.) 

The  general  conclusion  from  these  experiments  was  this,  viz. 
"  The  centre  of  tension  and  the  centre  of  compression,  each  co- 
incided with  the  centre  of  gravity  of  its  respective  area :  and  the 
neutral  line,  which  divides  the  two,  is  so  situated  that  the  area  of 
tension  into  the  distance  of  its  centre  of  gravity  from  the  neu- 
tral axis  is  to  the  area  of  compression  into  the  distance  of 
its  centre  of  gravity  from  the  same  line,  in  a  constant  ratio  for  each 
distinct  species  of  wood,  but  approximating  in  all  towards  the  ratio 
of  three  to  one. 

(94.)  This  theorem  being  established,  says  Mr.  Barlow,  it  is  evi- 
dent that  we  may  thence,  without  any  specific  numbers  for  exhibit- 
ing the  actual  resistance  of  the  fibres,  compute  the  proportional 
strengths  of  differently  formed  beams ;  and  of  the  same  formed 
beams  in  different  positions ;  of  which  we  will  give  one  example 
by  way  of  illustration. 

Problem  I. — Let  a  square  beam  be  fixed  with  one  end  in  a  wall, 
first  in  a  direct  position,  viz.  with  its  sides  perpendicular  and  hori- 
zontal ;  and,  secondly,  with  its  diagonal  vertical  to  find  the  ratio  of 
its  strength  in  these  two  positions. 

Conceive  ABCD  (fig.  83)  to  denote  the  beam  in  its  first  position, 
EF  the  neutral  axis,  EABF  the  area  of  tension,  and  t  the  centre  of 
tension  or  centre  of  gravity  of  that  area,  EDCF  the  area  of  com- 
pression, c  its  centre  of  gravity,  and  G  tlie  centre  of  gravity  of  the 
whole  area  of  fracture,  and  the  same  letters  will  denote  the  similar 
quantities,  in  fig.  84,  which  represents  the  section  of  the  beam  in 
its  second  position. 


THE  STRENGTH  AND  STRESS  OF  BEAMS.  109 

Then,  by  the  preceding  theorem,  we  have  (fig.  83), 

area  AEBFxn?x3=«rea  EDCFxnc, 
«    and  in  fig.  84,  area  EBFxn^x3=«re«  EFCDAxwc  ; 
both  which,  from  the  property  of  the  centre  of  gravity  at  (p.  59), 
are  reducible  to 

(fig.  83,)  area  AEBFx2  nt^area  ABCDxn  G 
(fig.  84,)  area  EBF  x2  nt=area  ABCDxn  G. 
For  the  sake  of  simplifying  the  computation,  let  the  side  of  the 
square  =1,  n)^=x,  or  nt^kx,  then  nG  =  d  —  x  ;  the  area  AEBF 
=a^,  and  the  area  ABCD^l,  whence  our  first  equation  gives 
x^=k  —  oc,  or  x^-\-x^h.  ;  whence  x= — 5±\/3  =  -366,  which  de- 
notes both  the  depth  of  tension  Hn,  and  area  of  the  same  AEFB  ; 

consequently,  •366x—^r— ='066978,  the  numerical  expression  for 

the  resistance  to  tension,  on  which  depends  the  strength  of  the 
beam.  It  remains  now  to  compute  the  same  for  the  second  position 
of  the  beam,  as  in  fig.  84. 

Here  if  we  denote  wB  by  x,  nt=l  x,  and  the  area  EBF=a?^  also 
area  ABCD=1,  as  before;  whence  our  second  equation  becomes 
a'^^|x=2^/2  — a?,  or  a;»+fx=|v'2  =  1-0606. 
From  which  we  readily  obtain  a'=-578  nearly, 
jj2:_. 33408,  and  x^^Ks  ^=5  a:='=-06436,  which  is  the  numerical  va- 
lue of  the  tension  in  this  position  of  the  beam.  The  strength  of  the 
beam,  therefore,  in  the  latter  position  is  to  that  in  the  former  in  the 
ratio  of  -06436  :  -06697  ;  or  as  the  numbers  643  :  669  nearly,  which 
accords  with  experimental  results,  and,  in  a  similar  way,  may  the 
strengths  of  difl'erently  formed  beams  be  compared.  We  shaU  now 
consider  the  straining  effects  on  beams  differently  supported,  and 
loaded  by  weights  at  different  parts.  j;^ 

Problem  II. — A  beam  of  timber  AB  (fig.  85)  is  fixed  with  one 
end  in  a  wall,  and  loaded  with  a  weight  W  at  the  other  end ;  to 
determine  the  efficacy  of  this  Aveight  to  break  the  beam. 

At  the  place  where  the  beam  has  the  greatest  tendency  to  break, 
the  broken  piece  will  tend  to  turn  round  the  neutral  axis ;  if  the 
distance  of  this  axis  from  W  be  /,  then  AV  will  express  the  energy 
of  W  to  produce  this  eff'ect ;  but  AV  will  be  greatest  when  /  is 
greatest,  that  is  when  this  denotes  the  whole  length  of  the  project- 
ing beam  ;  hence  the  beam  will  tend  to  break  close  to  the  Avail,  and 
ZW  will  express  the  strain  there.  The  strain  varies  therefore  as  the 
length  of  the  beam,  W  being  the  same. 

Problem  III. — A  beam  rests  loosely  on  two  props.  A,  B  (fig.  86), 
and  is  loaded  at  a  given  point  C  by  a  given  weight  W  :  to  determine 
the  stress  at  C. 
K 


110  ELEMENTS    OF    STATICS. 

Of  course  the  tendency  to  I)reak  will  be  at  C,  and  we  may,  there- 
tore,  assimilate  this  to  the  preceding  case  by  conceiving  the  beam 
to  be  fixed  in  a  wall  up  to  C,  and  to  be  strained  by  a  force  equal  to 
the  pressure  upon  the  prop  at  the  other  end. 

Now  by  (36)  the  pressure  on  the  prop  A  is  expressed  by 

P=£1LW: 
AB 

this  then  being  the  force  that  strains  the  projecting  beam  AC,  its 

AC  .  CB„, 
energy  is  — —^ — W. 

If  the  load  be  in  the  middle  of  the  beam,  the  product  AC  .  CB 
becomes  the  greatest  possible,  being  the  square  of  half  the  length ; 
hence  a  beam  will  be  less  able  to  support  a  weight  at  its  middle 
than  if  it  be  placed  at  any  other  part.  The  strain  obviously  varies 
as  the  product  of  the  distances  of  the  weight  from  the  props. 

It  may  be  further  remarked,  that  when  the  weight  acts  at  the 

AC^ 

middle,  the  stress,  being  =— — — W=|  AB  .  W,  is  one-fourth  the 

stress  which  the  same  weight  W  would  produce  acting  at  the  ex- 
tremity of  a  projecting  beam  equal  to  AC,  or  it  is  equal  to  the  whole 
stress  on  a  projecting  beam  equal  to  AB,  the  weight  at  its  extremity 
being  =i  W,  or  a  projecting  beam  equal  to  half  AB  and  loaded  with 
i  W  at  its  end,  will  sufler  half  the  stress  at  the  wall. 

Problem  IV. — To  determine  the  stress  on  a  projecting  beam, 
and  on  a  beam  resting  on  props  when  the  weights  are  distributed 
uniformly  over  them. 

Let  AC  be  the  projecting  beam  and  w  its  weight,  including  the 
uniformioad  ;  this  weight  will  act  at  half  the  distance  AC  from  the 
wall,  and  therefore  the  stress  is  k  AC  .  w,  which  is  just  half  what 
it  would  be  if  w  were  placed  at  the  extremity. 

When  the  beam  rests  on  props  (fig.  87),  the  pressure  on  each 

prop  is  half  its  weight  W,  this,  therefore,  is  the  force  acting  at  P 

which  tends  to  fracture  the  beam  at  C  with  an  energy  expressed  by 

i  W  .  AC  ; 

but  there  is  another  force  exerted,  viz.  that  due  to  the  weight  w  of 

the  portion  CA,  and  which,  by  last  case,  opposes  the  former  with 

an  energ}'  expressed  hy  ^  w  .  AC  ;  hence   the  expression  for  the 

stress  on  C  must  be  i  (W  —  jv)  AC  :    to  eliminate  w  we  have,  on 

account  of  the  uniformity  of  the  beam, 

AC 
AB  :  AC  :  :  W  :  w=^W, 
Ao 

SO  that  the  expression  for  the  stress  will  be 


THE  STRENGTH  AND  STRESS  OF  BEAMS.  Ill 

AB^C-AC^  AC^BC 

^       '  AB  ^~2  AB      • 

Hence  in  this  case  also,  as  in  problem  III.,  the  stress  varies  as  the 
product  AC  .  BC. 

The  strain  at  the  middle  point  of  the  beam  is  §  W  .  AB,  just 
half  what  it  would  be  if  the  whole  load  were  placed  there,  (pro- 
blem III.) 

SCHOLIUM. 

(95.)  By  means  of  these  problems  it  will  be  easy  to  find  the  most 
economical  forms  for  beams,  either  projecting  or  supported  at  the 
ends,  so  that  they  may  in  no  part  possess  superfluous  strength, 
that  is,  that  the  strength  in  every  part  may  be  exactly  in  proportion 
to  the  stress  there.  Thus  if  a  projecting  beam  of  uniform  breadth 
is  to  support  a  weight  at  its  extremity,  it  will  be  equally  strong 
throughout  if  the  vertical  sides  are  in  the  form  of  a  parabola  (fig. 
88);  for,  by  (prob.  II.),  the  stress  varies  as  AC=a:',  and  the 
strength  of  the  beam,  being  (93)  as  the  breadth  into  the  square  of 
the  depth,  and  the  breadth  being  constant,  varies  simply  as  CD^= 
y^ ;  hence,  in  order  that  the  strength  and  stress  may  be  throughout 

in  a  constant  ratio,  AC  must  vary  as  CD^,  that  is,  — — ^=constant 

AC 

=a  or  y'^=ax,  the  equation  of  a  parabola.  But  the  shape  need  not 
necessarily  be  parabolic  in  order  to  insure  uniformity  of  strength, 
as  it  will  depend  in  a  measure  upon  the  nature  of  the  vertical  sec- 
tions :  thus,  if  these  sections  are  required  to  be  all  squares,  then 
the  breadth  and  depth  being  every  where  the  same,  the  strength 
will  vary  as  the  cube  of  the  depth,  and  hence  AC  should  vary  as 
CD',  that  is,  y^=ax,  the  equation  of  the  cubical  parabola,  which 
must  therefore  be  the  form  of  the  tapering  beam. 

Again,  if  the  depth  of  the  beam  is  to  be  constant,  then  AC  should 
vary  as  the  breadth,  and  therefore  the  upper  and  under  faces  of  the 
beam  will  be  triangles.  If  a  beam  supported  on  two  props  is  to  be 
uniformly  strong,  and  at  the  same  time  uniformly  broad,  it  will  be 
necessary  to  form  the  vertical  sides  elliptical,  for  the  breadth  being 
constant  the  strength  will  vary  as  the  square  of  the  depth,  and,  by 
(prob.  IV.)  the  stress  at  any  point  C  varies  as  AC  .  CB  ;  hence  the 
square  of  the  depth  at  C  must  be  in  a  constant  ratio  to  AC  .  CB, 
which  requires  that  D  (fig.  89,)  be  always  in  an  ellipse  whose  axis 
is  AB,  (Jnal.  Geom.  p.  122.) 

(96.)  It  may  be  moreover  remarked,  that,  by  means  of  the  fore- 
going expressions  for  the  strain  or  tendency  to  produce  fracture, 
combined  with  the  results  of  experiment,  we  may  determine  the 


112  ELEMENTS  OF  STATICS. 

actual  weight  which  any  sfiven  beam  will  support  in  given  circum- 
stancns.  Thus,  suppose  it  is  f<>und  by  experiment  that  a  beam  of 
breadth  h,  depth  h,  and  lenffth  /,  just  breaks  with  a  weight  w  at  its 
middle,  and  that  it  is  required  to  determine  what  weight  W  will  just 
break  another  beam  of  like  materials  whose  breadth  is  B,  depth  H, 
and  length  L.  In  each  case  the  tendency  to  resist  fracture  is  just 
balanced  by  the  tendency  to  produce  it ;  the  expressions,  therefore, 
for  these  two  tendencies  must  be  equal,  and  therefore  the  ratio  of 
the  tendencies  to  resist  fracture  in  the  two  beams  must  equal  the 
ratio  of  the  tendencies  to  produce  fracture.  Now  the  tendency  to 
resist  fracture  is  what  we  understand  by  the  strength  of  the  beam, 
and  the  tendency  to  produce  fracture  is  the  stress ;  hence,  equali- 
zing the  two  ratios  spoken  of,  we  have  (prob.  III.) 

B  .  H"     :!  L  .  W  B^.  H^   /.  w 

b.h^  ~  U.io    ■**  ~t.h'  '    L    ' 

the  weight  required.  It  is  obvious  that  from  the  same  equation  we 
may  deduce  the  length  L  when  W  is  given,  and  also  that  the  equa- 
tion remains  the  same  whether  W,  w  act  in  the  middle  or  at  the 
end  of  each  beam. 

The  expression  here  given  may  serve  to  compare  the  strength  of 
any  beam  in  a  model  with  that  of  the  corresponding  beam  in  the 
structure.  Thus,  suppose  the  beam  which  we  have  considered  to 
have  been  submitted  to  experiment  to  belong  to  the  model,  and  the 
other  to  be  the  corresponding  beam  in  the  structure  whose  like 
dimensions  are  n  times  those  of  the  former,  then  the  foregoing  ex- 
pression for  W  will  be  W=n^  iv,  which  will  be  the  greatest  pos- 
sible load  tiie  beam  in  the  structure  can  bear,  including  of  course  its 
own  weight.  Now  if  the  weights  alone  of  the  two  beams  are 
respectively  p  and  P,  then,  since  these  must  be  as  the  cubes  of  their 
like  dimensions,  we  must  have  P=n^j3,  consequently  the  beam  in 
the  structure  so  far  from  bearing  a  load,  will  but  just  support  its 
own  weight  if  we  make  it  so  large  that  n^  p^n^w,  that  is,  if  n= 

— :  we  see,  therefore,  how  erroneous  it  would  be  to  estimate  the 
P 

strength  of  a  large  beam  in  a  structure  by  that  of  a  similar  small 
beam  in  a  model,  regarding  only  the  comparative  dimensions  of 
each ;  for,  by  increasing  the  magnitude  of  the  large  beam,  without 
in  the  least  changing  the  relative  proportions  of  the  two,  we  should 
nevertheless  render  it  at  length  too  large  to  support  even  its  own 
weight,  although  the  model  agreeably  to  which  it  has  been  formed, 
might  be  able  to  support  a  load  many  times  its  own  weight.  There 
is,  therefore,  necessarily  a  limit  to  the  magnitude  of  all  structures, 
even  indeed  to  the  magnitude  of  the  animal  structure,  and  to  trees, 
beyond  which  limit  they  would  be  unable  to  support  their  own 


81  A-  82 


83 

H 


84- 


^^m 


S3 


89 


"^~W1lJ|_ 


.It 


■H^''ilijlijr 


THE  STRENGTH  AND  STRESS  OF  BEAMS.  113 

weight ;  we  accordingly  find  men  of  enormous  magnitude,  as  O'Brien 
"  the  celebrated  Irish  Giant,"  to  be  so  weak  that  they  are  scarcely 
able  to  walk  about. 

In  connexion  with  these  remarks  may  be  mentioned  the  curious 
question,  proposed  by  Mr.  Emerson,  among  the  mechanical  pro- 
blems annexed  to  his  algebra ;  the  question  is  this  :  Supposing, 
with  Borelli,  that  a  strong  man  can  bear  but261bs.  at  arm's  end,  and 
that  the  weight  of  his  whole  arm  is  equivalent  to  41bs.  at  arm's  end  ; 
from  the  length  of  his  arm  being  given,  to  find  the  dimensions  of  that 
man's  arm  that  can  bear  no  more  than  its  own  weight. 

This  problem  is  immediately  solved  by  means  of  the  relation  n= 

w 

— ,  deduced  above,  w  representing  here  the  weight  26+4  or  30lbs., 

the  weight  of  the  common  man's  arm  and  load,  and  p  representing 
the  weight  41bs.  of  his  arm  alone,  so  that  n=7  5  :  this,  therefore, 
is  the  number  of  times  any  dimension  of  the  large  man's  arm  must 
contain  the  corresponding  dimension  of  the  common  man's  arm ; 
let  us  then  suppose  the  common  man's  arm  to  be  a  yard  long,  the 
length  of  the  other  man's  arm,  to  just  support  itself,  must  be  7^ 
yards,  and,  as  the  body  is,  in  well  proportioned  persons,  about  tAviee 
as  long  as  the  arm,  we  therefore  conclude  that  a  man  upwards  of  15 
yards  high  would  not  be  able  to  stretch  out  his  arm. 

Problem  IV. — To  determine  the  relative  strengths  of  beams 
loaded  in  the  middle  Avhen  their  ends  are  loosely  supported,  and 
when  they  are  firmly  fixed  in  two  vertical  walls. 

When  a  beam  is  loosely  supported  and  acted  upon  by  a  weight 
at  the  middle,  this  iveight  is  equally  divided  between  the  two  props, 
but  at  these  points  there  is  no  strain  ;  when,  on  the  contrary,  the 
ends  of  the  beam  are  firmly  fixed  in  immovable  walls,  then  it  is  the 
strain  on  the  middle  which  is  equally  divided  between  the  two  ex- 
tremities ;  that  is  to  say,  the  fibres  in  the  section  at  each  wall  are 
strained  half  as  much  as  those  at  the  middle  section.  The  whole 
of  the  weight,  therefore,  is  not  expended  here,  as  in  the  former 
case,  in  straining  the  middle  of  the  beam,  but  a  portion  is  employed 
in  straining  each  end  half  as  much.  Now,  whatever  weight  strains 
the  middle,  |  of  this  will  (by  prob.  III.)  strain  each  section  at  the 
wall  half  as  much ;  hence,  if  we  represent  that  part  of  the  weight 
which  strains  the  middle  only,  by  4,  the  part  which  strains  the 
ends  will  be  2,  and  therefore  the  Avhole  straining  weight  will  be  6, 
so  that  the  weight  6  will  produce  no  more  stress  on  the  middle  of 
the  beam  thus  fixed,  than  the  weight  4  when  the  ends  rest  loosely 
on  props  ;  hence  the  relative  strengths  of  fixed  and  loose  beams  are 
as  6  to  4  or  as  3  to  2,  Avhich  relation  Mr.  Barlotv  has  verified  by 
experiment. 

k2  15 


114  ELEMENTS  OF  STATICS. 

It  may  be  observed,  that  the  weiglit  8  uniformly  distributed  over 
the  beam  would  produce  the  same  strain  in  the  middle  as  the  weight 
4  applied  there  (prob.  IV),  and  the  weight  4  uniformly  distributed 
will  strain  the  ends  as  much  as  the  weight  2  applied  to  the  middle : 
hence,  if  the  load  be  uniformly  distributed  over  the  fixed  beam,  it 
will  be  no  more  strained  with  the  weight  12  thus  disposed,  than 
with  the  weight  6  acting  at  the  middle,  so  that  here,  as  in  the  other 
cases,  the  efliciency  of  the  beam  is  doubled  by  spreading  the  weight 
uniformly  over  it. 

SCHOLIUM. 

(97.)  The  result  of  the  preceding  investigation,  although  con- 
firmed by  Mr.  Barlow's  experiments,  differ  materially  from  tlie  con- 
clusions deduced  by  other  philosophers,  as  Girard,  Emerson,  and 
Jiobison,  who  find  the  comparative  strengths  of  supported  and  fixed 
beams  to  be  as  1  to  2,  and  not  as  2  to  3.  Emerson's  reasoning  on 
this  point  is  as  follows : 

Suppose  DA=AC  (fig.  90,)  and  BE=BC,  and  let  P  be  the 
weight  which  would  break  the  beam  when  resting  on  A  and  B. 
Suppose  the  beam  cut  through  at  C,  and  let  5  P  be  laid  upon  D, 
whilst  I  P  remains  at  C  ;  then  the  pressure  at  A  will  be  =P,  there- 
fore the  beam  will  also  break  at  A  having  the  same  stress  there  as 
it  had  at  C.  For  the  same  reason,  if  ^  P  be  applied  to  E,  CE  will 
break  at  B.  Consequently,  if  2  P  be  applied  to  C,  the  beam  being 
whole,  and  the  ends  D,  E  fixed,  the  beam  will  break  at  A,  C,  and 
B ;  and,  therefore,  bears  twice  the  weight,  or  2  P  at  C,  before  it 
breaks. 

Now  the  foregoing  reasoning  appears  to  assume,  that  before  the 
beam  can  break  at  C,  the  strain  on  A  and  on  B  must  be  sufficient 
to  break  the  beam  at  those  points  also ;  yet  it  is  shown  that  the 
beam  will  break  simultaneously  at  these  three  points,  if,  besides  the 
weight  P  acting  atC,  the  points  D,  C,  E,  be  each  loaded  with 
iP;  hence,  to  enable  the  middle  point  C  to  yield  to  the  pressure  of 
P,  it  is  only  necessary  that  the  fibres  at  A  and  at  B  be  half  as  much 
strained  as  they  are  by  the  influence  of  i  P  acting  at  D,  C,  and  E ; 
because  but  half  the  strain  of  the  fibres  at  A  are,  in  virtue  of  this 
influence,  in  the  direction  of  AC,  the  other  half  being  in  the  direc- 
tion of  AD  ;  and,  in  like  manner,  but  half  the  strain  at  B  is  in  the 
direction  of  BC,  so  that  if,  in  addition  to  P  acting  at  C,  as  much 
more  weight  is  added  as  will  produce  these  half  strains,  the  parts 
AC,  BC  will  be  deflected  sufficiently  for  the  beam  to  break  at  C  ; 
we  have,  therefore,  to  add  to  P  only  half  as  much  as  would  produce 
the  whole  strains  at  A  and  B,  that  is,  instead  of  P  we  should  add 
^P, making  the  whole  breaking  load  ^  P,  which  is  the  same  result 


THE    STRENGTH    AND    STRESS    OF    BEAMS.  115 

as  before  obtained,  and  the  coiTectness  of  which  Mr.  Barlow's  ex- 
periments confirm. 

Mr.  Barlow  remarks  on  this  subject,  "  in  every  experiment  that  I 
made  after  the  complete  fracture  in  the  middle,  the  two  fragments 
had  been  so  little  strained  at  the  points  of  fixing,  that  they  soon  after 
recovered  their  correct  rectilinear  form ;"  and,  in  order  to  show  the 
foundation  of  the  error  which  all  theorists  have  made,  in  assuming 
that  the  fixed  beam  would  break  simultaneously  in  the  middle  and  at 
the  walls,  he  further  adds,  "  If  the  beam  instead  of  being  fixed  at 
each  end  were  merely  rested  on  two  props,  and  extended  beyond 
them  on  each  side  equal  to  half  their  distance,  and  if  weights  w,  w' 
(fig.  91,)  were  suspended  from  these  latter  points  each  equal  to  one 
fourth  the  weight  W,  then  this  would  be  double  of  that  which  would 
be  necessary  to  produce  the  fracture  in  the  common  case  ;  for,  divi- 
ding the  weight  W  into  four  equal  parts,  we  may  conceive  two  of  these 
parts  employed  in  producing  the  strain  or  fracture  at  E, and  one  of  each 
of  the  other  parts  as  acting  in  opposition  to  iv  and  iv' ,  and  by  these 
means  tending  to  produce  fractures  at  F  and  F'.' 

"  This  is  the  case  which  has  been  erroneously  confounded  with 
the  former,  but  the  distinction  between  them  is  sufficiently  obvious  ; 
because,  here  the  tension  of  the  fibres,  in  the  places  where  the  strains 
are  excited,  are  all  equal ;  whereas  in  the  former  the  middle  one  was 
double  of  each  of  the  other  two."* 

Venturoli,  in  his  valuable  book  on  Mechanics,  says,  in  the  words 
of  Dr.  Creswell's  translation,  "  The  beam  would  sustain  a  load  con- 
siderably greater,  if,  instead  of  being  simply  placed  upon  two  props, 
it  were  immoveably  fixed  in  stone-work  at  both  its  extremities. 
For,  it  that  case,  it  cannot  break  unless  le  gives  way  in  three  places 
at  the  same  tirae."t 

Problem  V.  — To  determine  the  dimensions  of  the  strongest  rect- 
angular beam  that  can  be  cut  out  of  a  given  cylindrical  tree. 

Let  r  be  the  radius  of  the  base  of  the  cylinder,  and  x  and  y  the 
breadth  and  depth  of  the  required  beam,  then,  as  the  strength  varies 
as  xy^,  this  quantity  must  be  a  maximum  ;  hence 

Also,  as  the  diagonal  of  the  rectangle  is  equal  to  the  diameter  of  the 
circle,  we  have 

^•+!,.=4r».-.x+3,|=0.-.|  =  -^; 

•  Essay  on  the  Strength  and  Stress  of  Timber,  third  ed.  p.  149. 
f  VenturoWs  Mechanics,  Part  11.,  p.  60 . 


116  ELEMENTS   OF   STATICS. 

2  x" 

hence,  by  substitution,  y =0  .-.  3/^=2  x'=4  r' — x* 

2  r 
.-.  3  a:''=4  r^  .-.  x=— —  ,  y=2r  v/|,  the  dimensions  required. 

For  further  information  on  the  subject  of  this  chapter,  anil  more 
especially  for  an  account  of  the  various  experiments  that  have  hither- 
to been  made  to  determine  the  strength  of  materials,  the  student  is 
referred  to  Professor  Gregory's  valuable  Treatise  of  Mechanics ;  to 
the  second  volume  of  Sir  David  Brewster's  edition  of  Ferguson's 
Lectures ;  to  Mr.  Barlow's  work  on  the  Strength  and  Stress  of 
Timber,  as  also  to  his  treatise  on  Mechanics  in  the  Encyclopaedia 
Metropolitana ;  to  Part  II.  of  Creswell's  translation  of  Venturoli ; 
and  lasdy,  to  Professor  Leslie's  instructive  volume  on  the  Elements 
of  Natural  Philosophy. 

Perhaps  we  ought  to  remark  before  closing  this  chapter,  that  in 
all  the  foregoing  investigations  on  tlie  stress  of  beams,  we  have  not 
taken  into  account  the  deflection  from  the  horizontal  line  which  the 
force  produces  before  it  actually  breaks  the  beam.  By  reason  of 
this  deflection  the  energy  of  the  breaking  force  is  not,  strictly  speak- 
ing, expressed  by  the  intensity  of  the  force  multiplied  by  its  distance 
measured  along  the  beam  from  the  section  of  fracture,  but  by  the 
intensity  into  the  perpendicular  distance  of  the  fracture  from  its  di- 
rection ;  this  perpendicular  distance  is  equal  to  the  former  distance 
multiplied  by  the  cosine  of  the  angle  of  deflection,  and  therefore,  by 
introducir\g  this  cosine  as  a  factor  into  all  the  foregoing  expressions 
into  which  the  moments  of  the  straining  forces  enter,  they  will  be- 
come rigorously  correct ;  but  except  in  very  long  beams,  or  in  very 
elastic  ones,  the  deflection  is  too  small  to  render  this  modification  of  ' 
much  consequence :  Mr.  Barlow,  however,  has  not  neglected  its 
influence  in  his  important  inquiries  on  this  subject. 


END    OF    THE    ELEMENTS    OF    STATICS. 


PART   11. 

ELEMENTS  OF  DYNAMICS. 


SECTION  I. 

ON   THE    RECTILINEAR   MOTION    OF    A    FREE    POINT. 

(98.)  Having  considered  the  general  theory  of  equilibrating' 
forces,  we  come  now  to  Dynamics, the  second  principal  division  of  the 
science  of  Mechanics,  and  which  comprehends  the  theory  of  unba- 
lanced forces.  Dynamics,  therefore,  considers  bodies  in  a  state  of 
motion,  while  Statics  has  to  do  only  with  bodies  at  rest ;  in  this 
first  section  we  shall  confine  ourselves  to  the  consideration  of  recti- 
linear motion  only,  but  in  the  opening  chapter  we  shall  lay  down  a 
kw  general  and  fundamental  principles  which  always  hold,  whatever 
be  the  path  of  the  moving  point,  and  which,  in  fact,  will  be  found  to 
comprise  the  whole  theory  of  its  motion. 


CHAPTER  I. 

ON  THE  FUNDAMENTAL  EQUATIONS  OF  MOTION. 

(99.)  By  the  inertia  of  matter  is  meant  its  incapability  of  altering 
the  state  into  which  it  is  put  by  any  external  cause,  whether  that 
state  be  rest  or  motion. 

It  is  manifest  that  if  a  body  at  rest  receive  an  impulse  in  any  di- 
rection,* it  will,  if  entirely  at  liberty  to  obey  that  impulse,  move  in 
that  direction,  and  with  a  uniform  rate  of  motion  ;  for  as  we  suppose 
the  body  to  be  entirely  uninfluenced  by  any  other  cause,  and  since  it 
is  incapable  of  exertion  itself,  it  is  plain  that  for  whatever  reason  we 
could  suppose  the  motion  to  slacken  at  any  point  of  its  path,  for 
the  same  reason  we  might  suppose  the  motion  to  quicken.  The 
body  will,  therefore,  continually  move  at  a  uniform  rate  in  the  di- 
rection impressed  upon  it,  that  is,  if  notiiing  extraneous  interferes 
with  its  motion. 

*  The  body  is  here  considered  as  a  single  point,  or  else  as  recei^'ing  its  im- 
pulse towards  the  centre  of  gravity,  so  that  no  rotation  is  impressed  on  it. 

117 


118  ELEMENTS  OF  DYNAMICS. 

(100.)  Wc  have  just  spoken  of  the  rate  of  a  body's  motion  :  we 
estimate  this,  when  the  motion  is  uniform,  by  the  space  the  body 
passes  over  in  some  determinate  portion  of  time,  as  in  one  second, 
which  indeed  is  the  portion  generally  assumed  for  the  unit  of  time  ; 
so  that  when  we  observe  a  moving  body  to  pass  uniformly  over  ten 
feet  every  second  of  time,  we  express  the  rate  of  its  motion  I)y  say- 
ing that  it  moves  with  a  velocity  of  ten  feet,  or,  for  greater  brevity, 
that  its  velocity  is  ten  feet,  and  this  is  what  we  are  to  understand  by 
the  equation  v  =  \0  feet,  space  being  taken  as  the  measure,  or  repre- 
sentative, of  velocity. 

Suppose  now  that  t  represents,  not  the  time,  but  an  abstract  num- 
ber expressing  the  number  of  seconds  elapsed  sinc«  the  commence- 
ment of  the  uniform  motion,  and  let  s  denote  the  corresponding  space 
passed  over  by  the  body,  then  we  obviously  have  the  three  equations 

*  s 

v=—,  s=tv,  t=—, 
t  V 

so  that  any  two  of  the  three  quantities  s,  t,  v,  being  known,  we 
may  immediately  find  the  third. 

But  if  t"  is  not  reckoned  from  the  commencement  of  motion,  but 
only  after  a  certain  space  s'  has  been  described,  then,  s  being  the 
whole  space  gone  over  from  the  commencement,  the  three  equa- 
tions will  be 

s  —  s'  s  —  s' 

v= — - — ,  s=s  -^vt,  t=- 


t  V 

These  equations,  or  indeed  any  one  of  them,  comprehend  the 
whole  theory  of  the  motion  of  a  body  acted  on  by  a  single  impulse, 
or  influenced  by  any  cause  which  produces  uniform  motion.  We 
shall  give  an  instance  of  their  application. 

Two  bodies  a,  b  (fig.  92),  animated  by  the  velocities  v,  v'  set  out 
simultaneously  from  the  points  A,  B,  and  move  in  the  same  direc- 
tion AC  ;  to  determine  the  time  of  their  coming  together. 

Suppose  they  come  together  at  the  point  C,  then 

AC=r/,  BC  =  y'  /,  that  is,  calling  AC,  s,  and  AB,  s' 

s' 

s=vt,  s  —  s'=v  t  .wt — s  =v  t  .'.t= -, 

V  —  V 

that  is,  the  abstract  number  expressing  the  units  of  time  will  be  that 
which  arises  from  dividing  the  space  between  the  points  of  starting 
by  the  difference  of  the  spaces  denoting  the  velocities. 

It  may  be  remarked  here,  that  whatever  be  the  nature  of  the  in- 
fluence which  produces  uniform  motion,  and  which  we  have  above 
called  an  impulse,  we  have  a  right  to  conclude  that  its  effect  will 
be  proportional  to  its  intensity ;  in  other  words,  that  such  in- 
fluences, acting  on  the  same  body,  or  on  equal  bodies,  are  propor- 
tional to  the  velocities  they  produce 


FUNDAMENTAL   EQUATIONS    OF    MOTION.  119 

For  if  a  body  receive  a  certain  velocity  in  consequence  of  a  cer- 
tain impulse,  it  ought  obviously  to  acquire  double  that  velocity  if  at 
any  point  of  its  path  that  impulse  be  repeated  in  the  same  direc- 
tion, but  if  this  second  impulse  take  place  at  that  point  from  vi^hich 
the  body  set  out,  it  must  unite  with  the  first  impulse,  so  that  the 
consequence  of  a  double  intensity  of  impulse  will  be  a  double  ve- 
locity in  the  body,  and,  in  like  manner,  a  triple  intensity  will  pro- 
duce a  triple  velocity,  and  so  on. 

(101.)  Let  us  now  consider  the  circumstances  of  variable  mo- 
tion, and  let  us  first  ascertain  the  expression  for  the  velocity  of  a 
body  so  moving  at  any  epoch  t".  If  we  first  assume  that  the  ve- 
locity which  the  body  has  at  t"  continues  uniform  from  t"  to  t'\., 
then,  calling  t^  —  t,  A  /,  and  the  increment  of  the  space  or  s^  —  s, 

A  S 

A  5,  we  have  for  the  velocity  at  t"',v^= ,  however  small  A  t, 

A  t 

and  consequently  A  s,  which  depends  on  it,  may  be  ;  but  if  no  in- 
terval of  time  A  t"  exists  so  small,  during  which  the  velocity  does 
not  vary,  then  the  above  equation  is  true  only  when  A  t,  and  con- 
sequently A  s,  becomes  0 ;  hence,  by  the  principles  of  the  diffe- 
rs 
rential  calculus,  we  have  in  this  case  v=-r  •  Cl)>  which  is  there- 
at 

fore  a  general  expression  for  the  velocity  of  a  moving  body  at  any 
time  t"  however  its  motion  may  vary,  and  of  course  it  applies  also 
when  the  motion  is  uniform,  for  then 

AS      ds 

r= =-— -^  constant ....  (2). 

At      dt  ^  ^ 

(102.)  It  is  obvious  that  if  the  velocity  of  a  moving  body  con- 
tinually vary,  it  must  be  influenced  by  some  continuous  cause, 
however  this  cause  may  itself  vary  in  efficiency ;  for  from  the  in- 
stant the  cause  ceases  to  act,  that  instant  the  body  ceases  to  vary 
in  velocity  in  consequence  of  its  inertia.  We  call  the  cause  of 
variable  motion,  whatever  it  really  be,  force:  an  accelerative  force 
if  the  velocity  continually  increase,  and  a  retardive  force  if  the  ve- 
locity diminish.  We  shall,  in  our  general  reasonings,  consider  the 
force  as  accelerative,  because  in  order  to  adapt  our  conclusions  to 
retardive  forces,  it  will  be  necessary  merely  to  prefix  to  the  ex 
pression  for  F  the  negative  sign.  Let  us  now  investigate  this 
expression ;  and  first  we  must  remark,  that  as  the  effect  of  a  con- 
stant accelerative  force  is  obviously  to  generate  constant  increments 
of  velocity  in  equal  times,  if  we  agree  as  heretofore  to  represent 
causes  by  their  eflfects,  we  shall  obtain  the  expression  for  F  by  di- 
viding the  increment  of  the  velocity  by  the  units  in  the  increment 
of  the  time,  measured  from  any  epoch  t",  that  is, 


120  ELEMENTS    OF    DYNAMICS. 


F=^....(l): 


such  then  is  the  expression  for  a  constant  accelerative  force,  A  t" 
being  any  interval  of  time  from  /",  and  A  v  the  corresponding  aug- 
mentation of  velocity.  The  velocity  of  the  body  in  this  case  is  with 
propriety  called  a  uniformly  accelerated  velocity. 

(103.)  But  suppose  that  F  is  not  a  constant  force  ;  then  if  from 
any  epoch  t"  there  is  an  interval  A  /"  so  small  that  F  remains  un- 
changed throughout  it,  the  expression  just  given  will  in  that  case 
represent  the  intensity  of  the  force  acting  at  the  epoch  /",  and 
continuing  unabated  and  unaugmented  during  the  interval  A  t". 
If,  however,  choose  A  /"  as  small  as  we  will,  F  still  changes  during 
the  interval,  then  Ave  shall  express  this  fact  by  saying  that  the  in- 
terval A/",  during  which  F  remains  constant,  is  0 ;  hence,  for  a 

dv 
continually  varying  force,  the  expression  is  F=-^  (2),  and  this  may 

be  regarded  as  a  general  expression  for  the  accelerative  force  whether 

it  be  constant  or  variable,  for  when  it  is  constant 

x^      ^v      dv  ,„. 

F= =-r-=constant ....  (3). 

At      dt  ^  -^ 

We  may  give  a  different  form  to  the  general  expression  for  F,  for 

ds      „     dU 
smce  i»=^- .'.  F=-r— . . . .  (4). 

dt  df*  ^  ^ 

We  have  seen  (equa.  1,)  that  the  expression  for  F  at  any  epoch 
t"  is  equal  to  the  increment  of  the  velocity  that  tvould  he  generated 
in  any  number  of  seconds  after  that  epoch  (if  F  were  thence  to 
cease  to  vary,)  divided  by  that  number  ;  that  is,  F,  estimated  at  any 
epoch  t" ,  is  equal  to  the  increment  of  velocity  that  would  be  gene- 
rated by  that  force  constantly  acting  during  one  second.  But  the 
velocity  of  a  moving  body  at  any  epoch  is  measured  by  the  space 
it  would  pass  over  in  the  succeeding  second,  if  its  motion  were 
thence  to  become  uniform ;  hence  the  force  acting  upon  a  moving 
body  at  any  epoch  t",  is  measured  by  the  space  the  body  would 
pass  over  in  the  2d  second  of  time  after  t",  provided  it  were  to  pro- 
ceed during  that  second  with  the  increment  of  the  velocity  generated 
during  the  1st  second. 

It  thus  appears  that  both  velocity  and  force  may  be  measured  by 
space,  and  therefore  that  in  every  dynamical  inquiry,  where  the 
mass  is  not  considered,  the  only  concrete  quantity  concerned  is 
space,  for,  as  before  observed,  t  denotes  an  abstract  number,  viz.  the 
number  of  units  or  seconds  in  the  time  t". 

(104.)  It  should  be  remarked  here,  that  the  forces  of  which  we 
have  just  spoken  are  in  no  respect  influences  of  a  different  kind 
from  those  considered  in  statics  ;  they  merely  manifest  themsehes 


FUNDAMENTAL   EQUATIONS   OF    MOTION.  121 

differently  by  producing  different  effects,  and  it  is  to  the  effects  only 
that  we  look  in  estimating  these  influences.  The  statical  effect  of 
a  force  applied  to  a  body  is  pressure  or  weight,  and  we  accordingly 
represent  the  force,  in  statics,  by  pressure  or  weight.  The  dyna- 
mical effect  of  the  same  force  is  accelerated  velocity,  and  ac- 
cordingly we  represent  the  force  by  velocity  ;  or,  since  space  mea- 
sures velocity,  we  represent  it  by  space.  These  different  modes 
of  estimating  the  same  force,  therefore,  naturally  present  them- 
selves upon  observing  their  effects  ;  but,  for  all  the  purposes  of  com- 
parison, it  matters  not,  as  was  observed  in  Statics  (4),  by  what  we 
represent  the  efficiency  of  any  force,  taking  care  only  always  to 
keep  up  the  proportion  between  the  forces  and  their  representative 
quantities.  Thus  there  would  be  no  impropriety,  if  there  were  no 
inconvenience,  in  representing  an  accelerative  force  by  a  weight, 
provided  we  always  proportioned  the  weight  to  the  efficiency  of  the 
force  ;  and  this  leads  us  to  a  remark  of  some  importance,  viz.  that  the 
pressure  or  weight  produced  by  the  action^  of  a  force  on  anybody, 
is  to  the  pressure  or  weight  produced  by  the  action  of  any  other 
force  on  the  same  body,  as  the  acceleration  producedby  the  former 
force  is  to  the  acceleration  produced  by  the  latter :  for  it  is  plain 
that  the  ratio  of  the  two  forces  must  be  the  same  abstract  number 
however  they  are  represented  ;  so  that  if  we  know  the  two  pressures 
or  the  two  weights  wliich  the  forces  are  fitted  to  produce,  and  also 
the  acceleration  which  one  is  fitted  to  produce,  we  know  also  the 
acceleration  which  the  other  is  fitted  to  produce. 

(105.)  In  ail  the  foregoing  investigations  it  should  be  remarked, 
that  we  have  put  entirely  out  of  consideration  the  nature  of  the  path 
which  the  moving  body  describes.  All  that  we  have  said  as  to  the 
velocity  of  a  body  regards  its  rate  of  motion  along  the  path,  whether 
straight  or  curved,  in  which  it  happens  to  move,  and  has  nothing  to 
do  with  the  manner  in  which  that  motion  has  been  produced ;  for 
however  it  moves,  and  by  whatever  agency,  the  same  velocity  is 
always  expressed  by  the  same  linear  space.  So  too  with  regard  to 
the  moving  influence  itself,  or  the  force ;  this  also  has  been  esti- 
mated without  any  reference  to  the  path  along  which  it  impels  the 
body ;  but  it  should  be  observed,  that  a  force,  when  commencing 
its  influence  on  a  body,  may  find  that  body  already  in  motion,  and, 
as  is  easy  to  conceive,  may  act  on  it  so  as  to  divert  it  from  its  ori- 
ginal path,  and  cause  it  to  describe  some  other  ;  in  such  a  case  the 
body  may  be  moving  under  the  influence  of  two  forces,  or  under  the 
influence  of  an  impulse  and  a  force  ;  but  still  there  must  exist,  or 
at  least  we  can  conceive,  some  single  force  which  if  immediately 
applied  to  the  body,  at  any  instant  of  time,  would  give  it  the  same 
motion  that  it  actually  has  at  that  instant  in  virtue  of  the  combined 
influences  alluded  to.  Now  it  must  be  remembered  that  it  is  this 
L  16 


122  ELEMENTS  OF  DYNAMICS. 

single  and  equivalent  force  which  F  represents  in  the   foregoing 

equations,  and  which,   when  its  intensity  is  the  same,  is  always 

measured  by  the  same  linear  space  or  length  of  path,  be  this  path 

whatever  it  may.     By  the  path  of  a  body,  urged  by  an  accelerative 

force,  is  meant  the  track  of  its  centre  of  gravity. 

Another  circumstance  of  importance  deserves  to  be  mentioned 

ds 
here,  viz.  that  the  general  expression  -j-forthe  velocity  at  any  point 

of  the  path  is  no  other  than  the  differential  coefficient  of  the  variable 
path  s  taken  relatively  to  the  independent  variable  /.  It  is  from  this 
circumstance,  as  we  shall  hereafter  see,  that  we  are  enabled  to  de- 
termine the  path  of  a  moving  body  from  knowing  its  velocity  at  any 
point  of  it  in  quantity  and  direction.  In  like  manner  the  general 
expression  for  the  force  is  the  differential  coefficient  of  the  velocity 
taken  relatively  to  the  same  independent  variable,  and  this  expres- 
sion combined  with  that  for  the  velocity,  leads,  as  we  shall  presently 

dv 
see,  (equa.  D,)  to  the  expression  F:=v-j-,  which  is  sufficient  to 

determine  the  force  which  influences  the  body,  when  we  know 
what  function  the  velocity  v  is  of  the  space  s. 

(106.)  It  will  be  expedient,  for  the  convenience  of  reference,  to 
collect  together  here  the  fundamental  equations  of  motion  now  esta- 
blished, introducing  such  slight  modifications  of  form  as  may  tend 
to  facilitate  their  practical  applications  in  our  future  inquiries,  and 
deducing  from  them  such  brief  inferences  as  may  be  of  more  espe- 
cial interest  or  importance.  It  will  be  best  to  keep  distinct  those 
equations  which  refer  to  constant  forces,  or  to  motion  uniformly  ac- 
celerated, from  those  which  refer  to  variable  forces,  or  to  motion 
not  uniformly  accelerated. 

I.  TFhen  the  accelerating  Force  is  constant. 

Referring  to  equation  (2)  we  have 

dv=F  dt  .-.  v=Yt+c  ....  (A). 

If  t  become  0  when  v  does,  that  is,  if  t  is  measured  from  the  com- 
mencement of  motion,  the  constant  c  vanishes,  and  we  infer  from 
this  expression  for  v,  that  the  velocities  acquired  in  any  times,  reck- 
oning from  the  commencement  of  motion,  are  proportional  to  the 
times  themselves. 

Introducing  the  value  v=Ft  in  the  equation  (4),  we  have 
ds=Ftdt .-.  s=d  Ft'>=h't  ....  (B), 
no  constant  being  added,  because  s  vanishes  with  t.     From  this 
equation  we  infer,  that  the  spaces  measured  from  the  commence- 
ment of  motion  are  proportional  to  the  squares  of  the  times.     We 


FUNDAMENTAL  EQUATIONS  OF  MOTION,  123 

may  further  remark  here,  that  if  the  acquired  velocity  r=F/  were 
to  continue  uniform  during  the  time  it',  the  space  passed  over  in 
that  time  would  be,  (100),  s=:iFt^;  hence  the  space  described 
from  the  commencement  of  motion  is  equal  to  that  which  would  be 
described  in  half  the  tirpe  by  the  body  moving  uniformly  with  the 
acquired  velocity. 

If  we  eliminate  t  by  means  of  the  equations  (A),  (B),  disregard- 
ing the  constant  c,  we  shall  have 

v^=2Fs  .-.  i;  =  v/"2Fs (C) ; 

showing  that  the  spaces  described  from  the  commencement  are  prO' 
portionals  to  the  squares  of  the  acquired  velocities. 

The  foregoing  equations  are,  obviously,  sufficient  to  determine 
any  two  of  the  quantities  F,  t,  s,  v,  when  the  other  two  are  given, 
the  time  being  supposed  to  be  reckoned  from  the  beginning  of  the 
motion.  But  when  this  is  not  the  case,  and  the  time  is  supposed 
to  commence  not  till  the  body  has  acquired  a  given  velocity  v^,  then 
regard  must  be  had  to  the  constant  in  (A)  ;  the  value  of  this  constant 
is  plainly  0=?;^,  because  by  hypothesis  v^  is  what  v  becomes  when 
t=0.  Equation  (A)  will,  therefore,  here  be  v=Ft-\-v^  ....  (A')  ; 
equation  (B)  will  be  8  =  5:  Ft^-\-v^t .  .  .  .  (B')  ;  and  by  eliminating 
t  from  these  two,  we  have  for  (C)  the  equation 

ij2=2  Fs+?;,2  .'.V  \/"2Fs+t>7.  .  .  .  (C). 

II.  When  the  accelerating  Force  is  variable. 

From  equations  (4)  and  (2)  we  have 

u=-Tr,  F=-Tr  .'.  —=-r-''.Fds=vdv 
at  dt        V      ds 

.:fFdsr-riv^....(D); 

which  equation  is  sufficient  to  determine  the  velocity  v  when  we 

know  what  function  the  force  F  is  of  the  space  s,  or  it  is  sufficient 

to  determine  this  function  when  we  know  the  functions.     From 

the  same  equation,  also, 


^=/f 


dv (E) ; 

which  makes  known  the  space  described  when  we  know  what  func- 
tions V  and  F  are  of  this  space. 

And,  lastly,  from  the  equation  [4)  t=  I  —  .  .  .  .  (F)  ;  which 

determines  the  numerical  value  of  t. 

Having  established  these  equations,  we  shall  now  proceed  to  ex- 
hibit their  practical  application,  more  especially  to  those  motions 
which  are  presented  to  us  in  nature. 


124  ELEMENTS    OF    DYNAMICS. 

CHAPTER  II. 

ON    THE    RECTILINEAR    MOTION    I'RODUCED    BY   A   CONSTANT    FORCE. 

(107.)  The  most  remarkable  and  important  instance  of  the  action 
of  a  constant  force  is  that  wliich  nature  presents  to  us  in  what  we 
have  called  gravity,  being  that  force,  in  virtue  of  which  all  bodies 
near  the  earth  fall  to  its  surface,  with  a  uniformly  accelerated  ve- 
locity, in  a  vertical  direction. 

Numerous  and  very  accurate  experiments  have  fully  established 
the  fact,  tliat  the  velocity  of  a  falling  body,  when  all  resistance  is 
removed,  is  uniformly  accelerated,  and  that  its  direction  is  that  of  a 
vertical  line,  or  a  normal,  to  the  earth's  surface  at  the  point  M'here 
it  falls.  Such  experiments,  liowever,  made  at  any  particular  place 
on  the  motions  of  bodies  falling  from  a  small  elevation,  are  not  suffi- 
cient to  warrant  the  conclusion  that  gravity  is  really  a  constant  force 
in  the  acceptation  in  which  we  use  the  expression.  All  that  we  can 
fairly  infer  from  them  is  that,  at  the  same  place,  and  within  the 
range  of  small  elevations,  no  sensible  variation  of  force  is  discovera- 
ble, and  that,  therefore,  within  the  limits  of  our  experiments,  at 
least,  gravity  may  be  considered  as  a  constant  force.  But  to  ascer- 
tain the  real  nature  of  gravity,  by  means  of  such  experiments  as 
these,  it  is  obvious  that  they  ought  to  be  repeated  in  various  parts  of 
the  earth,  and  at  great  elevations  as  well  as  small.  This  indeed  has 
accordingly  been  done,  and  it  has  been  always  found  that  a  heavy 
body  carried  to  the  summit  of  a  high  mountain  loses  part  of  its 
weight,  shewing,  therefore,  that  gravity  acts  with  less  intensity  at 
the  summit  than  at  the  base  of  the  mountain  ;*  and,  on  the  contrary, 
it  has  always  been  found  that  the  body  increases  its  weight  when  car- 
ried into  those  latitudes  which  are  nearer  to  the  centre  of  the  earth. 
These  results  of  observation  are  doubtless  sufficient  to  show  that 
gravity  is  not  a  constant  force,  and,  moreover,  that  its  variation  de- 
pends, in  some  way,  upon  the  distance  from  the  centre  at  which  it 
acts.  But  it  is  doubtful,  chiefly  on  account  of  the  comparatively 
small  elevations  attainable  by  man,  and  partly  on  account  of  the  im- 

•  At  an  elevation  of  a  mile  above  the  surface  of  the  earth,  the  intensity  of  gra- 
vity is  diminished part ;  and  a  pendulum  clock,  beating  seconds  at 

•'  1977-291     f      '  »^  .  b 

the  level  of  the  sea,  would  lose  21-898  seconds  a  day  at  this  altitude,  a  quantity 
not  to  be  overlooked.  Any  traveller,  having  leisure  and  the  proper  apparatus, 
might  try  the  experiment  in  the  barrack  on  Mont  Cenis,  or  at  the  Hospice  of  St. 
Bernard. — HerscheVs  Physical  Astronony.     Ency.  Met- 


ON    RECTILINEAR    MOTION.  125 

perfection  of  instruments,  whether  from  such  experiments  the  real 
law  of  the  variation  of  gravity  could  have  ever  been  safely  inferred. 
The  discovery  of  this  law,  as  well,  indeed,  as  of  that  which  retains 
the  planets  in  their  orbits,  was  in  fact  the  result,  not  of  experiment, 
but  of  conjecture  ;  but  then  it  was  the  conjecture  of  Newton. 

He  was  the  first  who  conceived  the  splendid  idea,  and  who  after- 
wards fully  verified  and  established  the  important  fact,  that  the  at- 
tractive force,  not  only  of  the  earth  but  of  every  body  in  the  solar 
system,  decreases  in  intensity  in  the  same  proportion  as  the  square 
of  the  distance  from  the  centre  of  the  attracting  body  increases. 
This,  therefore,  is  the  law  of  universal  gravitation,  and  which,  as 
Sir  John  Herschel  beautifully  observes,  governs  equally  "  the  fall  of 
a  leaf  and  the  precession  of  the  equinoxes." 

The  investigation  of  this  law  is  not  fitted  for  this  place  ;  it  belongs 
indeed,  to  Physical  Astronomy,  but  we  propose  to  touch  upon  it 
hereafter,  at  present  we  confine  our  attention  to  those  motions  which 
take  place  near  enough  to  the  surface  of  the  earth  to  render  the  vari- 
ation of  gravity  inappreciable.  We  shall  shortly  see  that  the  ex- 
pression for  tlie  force  of  gravity  at  the  earth's  surface  is  about  32 
feet,  and,  from  the  observation  in  the  note,  it  appears  that  at  a  mile 
above  the  surface  this  value  is  diminished  only  by  about  the  2000th 
part,  which  is  too  small  to  affect  sensibly  the  circumstances  of  the 
motion  of  a  falling  body  computed  on  the  hypothesis  that  the  force 
suffers  no  variation  at  all. 

On  the  vertical  Motion  of  heavy  Bodies. 

(108.)  Let  g  represent  the  force  of  gravity,  then,  for  the  space 
descended  by  a  heavy  body  in  t  seconds,  we  have  by  (B)  the  ex- 
pression, 8=1  gt^  ;  and  consequently,  the  space  descended  in  one 
second  is  s=lg.  Now  this  space  has  been  ascertained,  by  very 
accurate  experiments,  to  be  in  the  latitude  of  London  I6J3  feet, 
very  nearly  ;*  hence 

16^Vft.=i^.-.  5-=i=32ift.; 
this,  therefore,  is  tlie  expression  for  the  force  of  gravity  at  the  earth's 
surface,  and  in  vacuo. 

1 .  To  determine  the  space  through  which  a  heavy  body  will  de- 
scend in  four  seconds  at  the  latitude  of  London,  and  also  the  velocity 
it  will  acquire. 


*  From  the  most  recent  experiments  in  the  latitude  of  London,  the  value  oi  g 
is  found  to  be  193-14  inches,  which  is  rather  greater  than  32i  feet,  this  latter 
being  indeed  the  value  of  gravity  at  about  the  latitude  of  45°.  The  number  32^ 
is,  however,  still  retained  in  most  of  our  elementary  books,  and  will  serve  equally 
well  for  the  purposes  of  practical  illustration. 
l2 


126  ELEMENTS   OF    DYNAMICS. 

Using  g  for  F  the  expression  (B)  gives  for  the  space 
s=h  £'/»=! 6-r'jX4«=257^  ft. ; 
also  the  equation  (A^  gives  for  the  velocity  i;=g-/=32^x4  =  128|  ft. 

2.  To  determine  in  what  time  a  heavy  body  will  descend  400 
feet. 


From(B)/=    l-?i=^rp?  =  4' 
^    ^        ^  g        >V/  321 


76 

77 
hence  the  time  is  4^f  seconds. 

3.  If  a  body  be  projected  downwards,  with  a  velocity  of  30  feet, 
in  a  vertical  direction,  how  fir  will  it  fall  in  four  seconds? 

By  equation  (B')  s  =  k  gt^'+v^  /=  IBy^X  16+30x4  =  3771 
feet. 

4.  A  body  is  projected  vertically  upward  with  a  velocity  of  120 
feet,  how  high  will  it  ascend  in  3  seconds  ? 

Here  since  gravity  retards  the  motion  of  the  body,  it  must  be 
considered  as  negative,  and  we  have,  from  equation  (B') 

s=  —  i  ^^2+t),  f=  — 16^^X^^+120x3=2151  feet. 

5.  To  what  height  above  the  surface  of  the  earth  will  a  body 
ascend  which  is  projected  vertically  upward  with  a  velocity  of  100 
feet? 

It  will,  obviously,  ascend  to  the  same  height  that  it  must  fall 
from,  to  acquire  a  velocity  of  100  feet ;  hence,  from  equation  (A), 

100 

^=5-^  •••^=373-1- =3 -11; 

and  from  equation  (B),  s=^  vt=\bbk  feet. 

7.  With  what  velocity  must  a  body  be  projected  to  reach  a  height 
of  579  feet  ? 

From  equation  (C)  i>  =  ^5 2  ^s|  =  ^/ 5  641x579 1  =  193  feet. 

8.  With  what  velocity  must  a  body  be  projected  downwards  from 
the  top  of  a  tower,  whose  height  is  150  feet,  so  that  it  may  arrive  at 
the  bottom  in  two  seconds? 

Calling  the   velocity  v^,  equation  (B')  gives  s=5  gt^-\-t\t 

s                   150 
.•.Vj= lgt=—^ 16yVx2=42|  feet. 

9  Suppose  a  body  is  let  fall  from  a  height  of  300  feet,  and  that 
two  seconds  afterwards  another  body  is  let  fall  from  a  height  of  200 
feet,  in  what  time  will  the  former  overtake  the  latter  ? 

Let  us  suppose  that  the  second  body  will  have  been  in  motion  x 
seconds  when  the  first  -overtakes  it,  then  the  first  will  have  been  in 
motion  .r  +  2  seconds ;  consequently,  the  space  described  by  these- 
''ond  will  be  s=^\  gt^=^\Q^:^^^;  and,  therefore,  the  space  described 


ON    RECTILINEAR   MOTION.  127 

by  the  first  must  be  16  ^Va^^+lOO,  but  this  space  is  also 

5=1  ^/==16yWa?+2)^  consequently,  16^2^?^+ 100  =16x^(^^+2)^ 

107 
.-.  100=641  a:+64^.-.a:=-j^; 

hence  they  will  meet  if  |^  of  a  second. 

10.  How  far  must  a  body  fall  to  acquire  a  velocity  of  90  feet  ? 

Ans.  125-9  feet. 

11.  What  space  was  described  in  the  last  second  by  a  body  which 
had  fallen  7  seconds  ?  Arts.  209^^  feet. 

12.  With  what  velocity  must  a  body  be  projected  into  a  well  350 
feet  deep,  that  it  may  arrive  at  the  bottom  in  4  seconds  ? 

Ans.  231  feet. 

0)1  the  Motion  of  Bodies  along  inclined  Planes. 

(109.)  When  a  body  is  placed  on  an  inclined  plane  the  force  of 
gravity  produces  a  certain  pressure,  represented  by  the  weight  of 
the  body :  if  we  resolve  this  vertical  pressure  P  in  two  directions, 
the  one  along  the  plane,  and  the  other  perpendicular  to  it,  the  former 
component  will  be  P  sin.  ^,  taking  i  for  the  inclination  of  the  plane 
to  the  horizon,  and,  to  prevent  the  body  from  moving  down,  this  is 
the  force  or  pressure  which  must  be  counterbalanced.  As,  therefore, 
P  represents  the  force  of  gravity  in  the  vertical  direction,  and  P  sin. 
i  the  force  in  the  direction  of  the  plane,  and  moreover,  as  g  repre- 
sents the  vertical  acceleration,  we  shall  have  for  the  acceleration 
down  the  plane,  (see  p.  121,)  P  :  P  sin.  i  ::  g  :  g  sin.  i  ;  hence 
the  body  is  urged  down  the  plane  by  the  constant  force, ^'=^sin.i; 

and,  therefore,  substituting  in  the  formulas  (106)  this  value  of  fif' 
for  F,  they  will  then  comprise  the  whole  theory  of  motion  down  an 
inclined  plane,  whether  the  body  have  an  initial  velocity  up  or  down 
the  plane  or  not. 

If  I  represent  the  length  of  the  plane,  and  h  its  height,  then 

Tt  h 

sin.i=-y;  hence  the  accelerating  force  is  g-j-  '•>  and,  therefore,  the  velo- 
city acquired  in  descending  down  the  whole  length  /,  that  is  in  de- 
scending through  the  space  s=/,  by  the  influence  of  this  force  must 
be  (C),  v=  v^  {'^S^\  ;  which  expression,  being  independent  of  /, 
shows  that  the  velocity  acquired  in  descending  down  all  planes  of 
the  same  height  is  equal  to  the  velocity  acquired  in  falling  through 
that  height. 

The  velocities  of  two  bodies,  the  one  falling  through  the  perpen- 
dicular height,  and  the  other  falling  through  the  length  of  the  plane, 
are  respectively  i;=g-^,  u'=g-/' sin.  i ;  but,  as  these  velocities  are 


128  ELEMENTS    OF    DYNAMICS. 

equal,  we  must  have  gt^gt'  s'ln.i  .'.  /=/' sin.  i;  so  that  the  time 
of  falling  through  the  height  is  to  the  time  of  Hilling  through  the 
length,  as  sin.  i  to  1.  But  if  we  wish  to  know  what  extent  of  length 
is  gone  through  by  the  one  body,  while  the  other  goes  through  the 
whole  height,  then  referring  to  the  expressions  for  the  spaces,  we 
have 

5=1.  gP=i  gn,  s'=^gt"'  sin.  i  ; 
and  these  also  are  to  each  other  as  1  to  sin.  i.  If,  therefore,  from  B 
(fig.  93)  we  draw  the  perpendicular  BD,  AD  will  be  the  length  gone 
through  by  one  body,  while  the  other  falls  through  the  height  AB, 
because  AB  :  AD  :  :  1  :  sin.  ABD=sin.  C=sin.  i.  If  we  draw  the 
vertical  DB'  and  BB'  perpendicular  to  DB,  then  the  time  of  falling 
through  DB'  would  equal  the  time  of  falling  through  DB,  but 
DB'  =  AB,  therefore  the  time  of  falling  through  AB  is  equal  to  the 
time  of  falling  through  either  of  the  inclined  planes  AD,  BB'.  Hence 
this  remarkable  property  of  the  circle,  viz.  :  If  from  the  extremities 
A,  B,  (fig.  94,)  of  the  vertical  diameter  AB,  cords  be  drawn,  a  body 
would  fail  through  either  of  them  in  the  same  time  that  it  would  fall 
through  the  vertical  diameter. 

(110.)  AVe  shall  now  add  an  example  or  two  of  motion  on  an  in- 
'•lined  plane. 

1.  The  length  of  an  inclined  plane  is  60  feet,  and  its  inclina- 
tion 30°,  what  velocity  would  a  body  acquire  in  falling  down  it  for 
2"  ? 

Substituting,  g  sin.  i,  for  F,  in  the  equation  (A),  we  have 
v=gt  sin.  z=32ix2x|=32i  feet. 

2.  How  long  would  a  body  be  in  falling  down  an  inclined  plani 
whose  length  is  100  feet,  and  inclination  60°  ? 

Substituting  g  sin.  i  for  F,  in  the  equation  (B),  we  have 


/  =     f : :=       I r = r-TTT-, ! ^TT  =  2*6    SBCOnds. 

\^gsm.i      \32ixlN/3      x/116^VX3v'3^ 

3.  If  a  body  be  projected  up  an  inclined  plane  whose  length  is 
ten  times  its  height,  with  a  velocity  of  30  feet,  in  what  time  will  the 
velocity  be  destroyed  ? 

The  time  is  necessarily  the  same  as  would  be  required  to  produce 
a  velocity  of  30  feet  in  a  body  falling  from  rest  down  the  same 

plane  ;  hence  making  the  substitution  of  ^  -^  for  g,  in  the  equa- 

*■       /AN  u        ,        •^^       30x10 

tion  (A),  we  have  t  =  — —=  ^  ~, — i=9-3  seconds. 
^    '  gh       321x1 

4.  A  body  is  projected  up  an  inclined  plane  whose  height  is  ^th 
of  its  length,  with  a  velocity  of  50  feet.  Find  its  place,  and  the 
velocity,  after  6"  have  elapsed. 


ON  RECTILINEAR  MOTION.  129^ 

Here  the  force  g  -j-  retards  the  motion  of  the  body,  and  must, 

therefore,  be  considered  as  negative  :  hence,  from  equation  (B'),  we 
have 

h  1 

s=v^  t  —  h  g-— T-i^=50x6  — 16  rVx  — X36=203^  feet, 

and  from  equation  (A),  v=  —  g—j-t=  —  32i  x-6-X6=  —  32J-  feet ; 

.*.  50  —  32i  =  17|-  feet,  the  velocity  required. 

5.  How  long  would  a  body  be  in  falling  down  an  inclined  plane 
whose  height  is  to  its  length  as  7  to  15,  to  acquire  a  velocity  of  20 
feet?  ^ns.  1*3  seconds. 

6.  Required  the  length  of  a  plane  whose  inclination  is  30°  that 
will  cause  a  body  let  go  at  the  top,  to  acquire  a  velocity  of  500 
feet  when  it  reaches  the  bottom.  ^ns.  7772  feet. 

It  should  be  remarked,  that  in  what  is  here  said  about  motion 
along  an  inclined  plane,  friction  is  entirely  disregarded ;  the  body 
being  supposed  to  slide  freely  down  the  plane  without  suffering  the 
least  impediment. 

(111.)  The  two  problems  following  are  added  as  a  further  illus- 
tration of  the  motions  of  bodies  under  different  modifications  of 
gravity,  and,  also,  as  an  additional  application  of  the  principle  stated 
at  (p.  121). 

Problem  I. — Two  weights  W,  W^,  connected  by  a  thread 
passing  over  a  small  pulley  C,  as  in  fig.  95,  are  placed  upon  the  two 
inclined  planes  CA,  CB ;  to  determine  the  circumstances  of  their 
motion. 

The  vertical  pressure  of  the  whole  mass,  produced  by  the  force 
of  gravity,  is  W  +  W^;  the  acceleration  which  would  be  produced 
by  the  same  force  is  g.  Again,  the  pressure  of  W  in  the  direction 
WA,  or  which  is  the  same  thing  the  tension  of  tlie  thread,  AV^  C, 
is  W  sin.  i  ;  also  the  pressure  of  W^,  in  the  direction  W^  B,  is 
Wj  sin.  ij ;  hence  the  system  must  move  in  virtue  of  the  difference 
of  these  two  pressures  and  to  find  with  what  acceleration  F  we  have 
(p.   121,) 

W  +  W, :  W,  sin.  i,-W  sin.  i  :  g  :  :  ^^ '' V,  + W  '"'"  '  ^=^  ' 
this,  therefore,  is  the  expression  for  the  accelerative  force  which 
urges  Wj  down  the  plane  CB,  and  which,  consequently,  draws  W 
up  the  plane  AC  ;  and,  therefore,  substituting  this  expression  in- 
stead of  F  in  the  equations  (A)  and  (B),  at  art.  (106),  we  have,  for 
the  velocity  acquired  and  space  passed  over  at  the  end  of  i  seconds, 
after  the  commencement  of  motion,  the  expressions 

17 


130  ELEMENTS  OF  DYNAMICS. 

W,  sin  i,  —  W  sin.  I  W,  sin. /,  —  W  sin.  {    ,„ 

v= — "■/:  s= — s:t' 

W,+  W  *   '  2(W,+  VV)         ^ 

If  the  two  planes  were  vertical,  then  the  problem  would  be  to  de- 
termine the  motion  when  the  two  weights  hang  vertically  at  the 
ends  of  a  tliread  passing  over  a  pulley  ;  since,  therefore,  in  this 
•ase,  sin.  i  and  sin.  i^  are  each  unity,  we  have 

F      W,  — W  W,  — W  AV,  — W 

If  only  one  of  tlie  planes  were  vertical,  the  problem  would  be  to 
determine  the  motion  when  one  weight  W^,  hanging  freely,  draws 
another  W  up  an  inclined  plane.     In  this  case  sin.  ii=l 
W,  — Wsin.  z 

w,+w     * 

If  one  of  the  planes  were  vertical  and  the  other  horizontal,  the  pro- 
blem would  be  to  determine  the  motion  when  Wj,  hanging  vertically, 
draws  W  along  a  horizontal  plane.     In  this  case  sin.  ?=0 

Problem  II. — A  given  weight  Wj  is  to  draw  another  given  weight 
W  up  an  inclined  plane  of  given  height  h  ;  required  the  length  /  of 
the  plane  in  order  that  the  time  of  ascent  may  be  the  least  possible. 

The  inclination  of  the  plane  being  represented  by  i,  as  usual,  we 

have  sin.  i=-t->  and  the  above  expression  for  F  may,  therefore,  be 
written 

hence,  by  equation  (B),  the  expression  for  s  or  the  length  /  is 
w   I W/i 


2     '  \(W,/— W/t)^' 


w^  /+w/ 

this  expresses  the  time  when  /  as  well  as  h  is  given.  To  deter 
mine,  therefore,  the  value  of  this  expression  when  a  minimum,  we 
must  put  the  first  differential  coefiicient  derived  from  it,  equal  to  0, 
/  being  the  independent  variable ;  or,  we  may  omit  the  radical,  as 
also  the  constant  factors  before  differentiating,  (Diff.  Calc.  p.  8,) 
and  we  shall  theri  only  have  to  make 

/«  W   /_W/t      W,        W/i 

wr/^^WA=™'"-  •••  —7^ =-7 jr=^^- 

W,     2W/i     ^       ,    2WA 


ON  RECTILINEAR  MOTION.  131 

On  the  Motions  of  Projectiles. 

(112.)  Altliougli  we  do  not  intend  to  consider  the  general  theory 
of  curvilinear  motion  in  the  present  section,  yet  it  will  be  advisable 
to  discuss  here  that  particular  case  of  it  which  we  observe  in  bodies 
when  projected  obliquely  into  space,  near  the  earth's  surface.  We 
know  that  every  body  so  projected  is  influenced  by  two  distinct 
causes,  viz.  the  primitive  impulsion  of  which  the  effect  is  to  give 
the  body  some  determinate  and  uniform  velocity  in  a  straight  line, 
and  the  force  of  gravity,  of  which  the  effect  is  continually  to  draw 
down  tlie  body  in  a  vertical  direction ;  these  verticals  tend  to  the 
earth's  centre,  but  throughout  the  path  of  a  projectile  they  may 
without  sensible  error  be  considered  as  parallel.  On  this  hypo- 
thesis, and  abstracting  for  the  present,  as  in  the  case  of  falling  bodies, 
from  the  resistance  of  the  air,  we  may  easily  determine  the  curve 
which  the  body  describes.  There  are,  indeed,  two  methods  of 
solving  very  readily  this  problem  ;  one  method  is  first  to  express 
by  means  of  horizontal  and  vertical  co-ordinates  the  equation  of  the 
straight  line  which  the  impulsion  would  compel  the  body  to  describe, 
if  gravity  did  not  act,  and  then  to  diminish  the  ordinate  y  by  the 
deflection  which  gravity  would  cause  for  the  time  t" .  Thus,  as- 
suming the  point  of  departure  as  the  origin  of  the  horizontal  and 
vertical  axes,  we  have,  for  the  initial  direction  of  the  body,  the 
equation 

y=ax ; 
but,  in  the  time  t"  gravity  diminishes  this  value  of  y  by  |  gf^, 
,•.  y=:ax  —  i  gt^  ....  (1). 

If  Vj  be  the  velocity  of  projection,  v^  wall  express  the  linear 
space  which  in  the  absence  of  gravity  the  body  would  pass  over  in 
1"  ;  hence  in  t"  it  would  pass  over  i\  t.  Now  the  action  of  gra- 
vity being  always  vertical,  it  is  obvious  that  this  force  cannot  at  all 
affect  the  horizontal  advance  of  the  moving  body,  so  that,  coitcs- 
ponding  to  any  time  t",  the  abscissa  will  be  the  same  whether 
gravity  act  or  not;  but,  from  what  has  just  been  said,  this  abscissa, 
in  the  absence  of  gravity,  is  v^t  cos.  d,  o  being  the  angle  of  eleva- 
tion of  the  piece  ;   hence 

a?=u    t  cos.  6  .'.  t= ....  (2). 

t^l    COS.   9 

Substituting  this  value  for  t,  in  the  equation  (1),  we  have,  for  the 
equation  of  the   path,  2/=tan.   0.  x .  (3);     which 

Z  V -^     COS.     B 

shows  that  the  path  of  the  projectile  is  a  parabola,  and  that  the 
rectangidar  axes  are  parallel  to  those  of  the  curve  (Anal.  Geom.  p. 
183),  so  that  the  vertex  of  the  parabola  is  the  highest  point  of  it. 


132  ELE>1EXTS  OF  DYNAMICS. 

Tf  h  denote  the  height  due  to  the  velocity  i',,  that  is  to  say,  the 
height  from  which  a  body  must  fall  vertically  to  acquire  this  velo- 
city, then  since  (C)  v^=^/\2.  gh\  llie  equation  may  be  written 

V=tan.  e  X -—r T—x^  ....  (4) 

^  Ah  cos.«  e  ^ 

The  other  method  of  obtaining  the  equation  of  the  path  to  which 
we  have  alluded  is  this.  Taking  the  same  origin  as  before,  let  the 
direction  of  projection  be  taken  for  the  axis  of  y,  and  a  vertical  line 
drawn  downwards  for  the  axis  of  x ;  then  v^  being  the  initial  velo- 
city, as  before,  we  have 

x  =  l  gt\  y=v,  t=fy\2  gh\   ....  (1); 
h  being  the  height  due  to  the  initial  velocity. 

Eliminating  f  we  get  1/^=4  hx  ....  (2) ;  the  equation  of  the 
parabolic  path,  and  from  which  it  appears  that  h  is  the  distance  of 
ihe  origin,  or  point  of  projection,  from  the  focus  of  the  parabola,* 
and  as  this  is  equal  to  the  distance  of  the  same  point  from  the  di- 
rectrix, it  follows  from  equation  (1)  that  the  velocity  at  any  point 
of  the  curve  is  equal  to  the  velocity  acquired  in  falling  vertically 
from  the  directrix  to  that  point.  Having  thus  determined  the  nature 
of  the  path  of  a  projectile,  we  shall  now  subjoin  a  few  general 
problems  arising  out  of  this  determination. 

Problem  I. — (113.)  To  determine  the  angle  of  elevation  9,  for 
which  the  range  AB  may  be  the  greatest  possible. 

The  general  expression  for  the  range  or  horizontal  distance  is 

the  value  of  x,  given  by  equation  (4)  for  y=0,  that  is,  it  is 

x=4  h  cos,''  B  tan.  0=2  h  sin.  2  0 (1)  ; 

and  as  this  expression  is  to  be  a  maximum,  we  must  have 

dx 

— =4  h  COS.  2  6=0. -.6=45° (2); 

(1 9 

which  gives  from  (1)  x=2  h  .  (2),  for  the  greatest  range. 

As  sin.  2  9=sin.  2  (90°  —  O),  it  follows  from  the  general  expres- 
sion (1)  for  the  range,  that  the  range  is  the  same  for  90°  —  9  as  for 
6,  that  is  the  ranges  are  the  same  whether  the  initial  direction  forms 
an  angle  below  the  line  of  45°  or  an  equal  angle  above  it,  (fig.  96.) 

Problem  II. — Knowing  the  range  of  a  shot  with  a  given  charge 
of  powder  and  a  given  elevation  of  the  piece,  to  determine  the  range 
at  any  otiier  elevation. 

Suppose  we  know  the  maximum  range  R,  or  that  due  to  the  ele- 
vation of  45°,  then  from  equation  (2),  above,  the  height  due  to  the 
velocity  of  projection,  is  h=t  R  ;  hence  this  is  the  value  of  h,  for  all 

•  Any  point  in  the  path  may,  obviously,  be  considered  as  the  point  of  projection. 


ON  RECTILINEAR  MOTION.  133 

elevations  with  the  same  charge.     Calling,  therefore,  the  range  due 

to  any  other  elevation  e,  r  the  expression  for  its  value,  will  be  r  = 

R  sin.  2  6. 

Thus  any  range  is  known  by  means  of  the  maximum  range.     Or 

if  we  know  any  range  r  corresponding  to  the   elevation  e,  then  to 

determine  the  range  r'  corresponding  to  another  elevation  9' ,  we 

have  the  two  equations  r=R  sin.  2  e,  ?''=R  sin.  2d', 

,.    .         _     ,  r'      sin.  2  e'         ,     sin.  2  e' 

to  eliminate  R ;  hence  — =—. — r —  .'.  r  =■—. — - —  r. 
r      sin.  2  e  sin.  2  0 

Problem  III. — Given  the  angle  of  elevation  and  the  initial  velo- 
city, to  determine  the  time  of  flight,  and  the  greatest  height  of  the 
projectile. 

Returning  to  equation  (1),  art.  (112),  we  have,  when  2/=0, 
I2  ax  _    I  2  Uni.  9.  X 

but,  by  problem  I.,  x=4h  cos.^  9  tan.  e  ;  hence  by  substitution 
t =2  sin.  9    I .  .  .   .(1);  which  expresses  the  time  of  flight. 

To  determine  the  greatest  height  above  the  horizontal  plane  we 
must  find  the  maximum  value  of  y,  from  equation  (4)  art.  (112), 
for  which  purpose  we  have  the  equation 

-^=tan.  0 — =0  ....  (2) 

dx  2  h  COS. 2  9  ^  ' 

.•.  x=2h  C0S.2  9  tan.  e=h  sin.  2  0  ....  (3) ; 

which,  by  equation  (1)  prob.  I.,  is  half  the  whole  range:  putting 

this  value  for  x  in  the  equation  of  the  curve,  we  have  for  y 

y=2  h  sin.®  6  —  h  sin.^  9—h  sin.^  9  .  .  .  .  (4), 

which  expresses  the  greatest  height. 

If  61=45°,  sin.''  9  =  h  .'•  y=i  h,  so  that  (prob.  I.),  the  greatest 

height  is  one-fourth  of  the  range. 

The  expression  (2)  denotes  the  tangent  of  the  angle  which  the 

curve  makes  with  a  horizontal  line  at  any  point  (a:,  y). 

Problem  IV. — Given  the  initial  direction  to  determine  the  velo- 
city, so  that  the  projectile  may  pass  through  a  given  point. 

Let  [x',  y')  be  the  given  point,  then  by  the  equation  of  the  curve 
(p.  131), 


s^x'^  X 

V  =tan.  9.  x' ^= .•,  v.= 


g- 


2v^^cos.^9  '  '    ^     COS.  0%  2  (tan.  e-  x' — y') 

Problem  V. — When  the  velocity  of  projection  is  given,  to  deter- 
mine the  direction  so  that  the  projectile  may  pass  through  a  given 
point. 

M 


134  ELEMENTS   OF    DYNAMICS. 

By  substituting  in  equation  4,  art.  (112),  sec. '9,  or  rather  1  + 

tan.'fl,  for 

1  ,  l+tan.=  e   ,„ 

;,— ,  we  have  7y'=tan.  9.  x' — ; x  " 

cos.*»  *^  4  A 


tan.''  9 tan.  9=. 1  ;  this  quadratic  solved 

x'  x"^ 


4  h  Ah  y' 

x'         '  x"^ 

for  Ian.  0  gives  tan.  9= ^-^ ; •  .    .  .    (1) ; 

so  that  there  are  two  difTerent  directions  whenever  the  problem  is 
possible,  except  when  'ih^=4lnj'  —  x"^,  or  (2h — y'y=x'"-\-y'^,  in 
which  case  there  is  but  one  direction,  but  when 

{2h—y'y>x'-'+y", 
the  problem  becomes  impossible  under  the  proposed  conditions. 

The  time  elapsed  from  the  instant  of  projection  till  the  projectile 
reaches  the  proposed  point  is,  by  equation  (1),  art.  (112), 
\2y'  +  2Um.9.x'\ 


'=4- 


Problem  VI. — To  determine  the  range  on  an  oblique  line  passing 
through  the  point  of  projection,  and  also  the  time  of  flight. 

Let  i  be  the  inclination  of  the  oblique  line  to  the  horizon,  then 
its  equation  is  ^'=tan.  i.  x'  .  .  .  .  (1);  combining  this  with  the 
equation  of  the  projectile  we  shall  obtain  the  abscissa  of  the  point, 
where  it  meets  this  line,  by  the  equation 

•      ,  ,  ^''^ 

tan.  t .  X  =tan.  9.  x ; -— 

4  A  COS.*  9 

•v     4  A  COS.  e  sin.  (e  —  i) 

.'.  x'=4h  COS.''  9  (tan.  e  — tan.  i)= ^-^^ -.  .  .(2); 

^  ^  cos.  t 

and,  consequently,  the  oblique  range  will  be 

x'        4h  COS.  9  sin.  (9 — -i)  ,„, 

r= := —7-^ .  .  .  .  (3)  ; 

COS.  I  COS.''  t 

and  the  time  may  be  found  from  equation  (2),  last  problem,  by  sub- 
stituting for  x'  and  y'  the  values  (1)  and  (2)  in  this.     This  substi- 

/tan.  0  —  tan.  i)  8 /j  COS.  e  sin.  (e  —  i). 
tution  gives  t=s/\- ■ ^ -\ 

^  *  g  COS.  I 

COS.   I  g 

Problem  VII. — To  determine  the  greatest  range  on  an  oblique 
plane,  and  the  greatest  height  above  it. 

The  angle  of  elevation,  which  belongs  to  the  greatest  range,  will 
be  that  which  renders  the  expression  (3),  last  problem,  a  maximum, 
or,  since  i  is  constant,  we  must  have 


'  ON  RECTILINEAR  MOTION.  135 

2  COS.  e  sin.  (9 — i)=max. 

=sin.  \d  +  {9  —  i)\ — sin.  {e  —  {o  —  i)] 

=sin.  (2  6  —  i)  —  sin.  i 

.-.  sin.  (2  9— z)=max.  .-.2  9  — i=90°  .-.  e  =  k  (90°+i). 

Putting,  therefore,  this  value  of  6  in  the  expression  (3)  for  the 

range,  we  have  for  the  maximum  range  R 

P_4  h  COS.  h  (9D°+0  sin.  |  (90°  —  i) 

cos.^  i 

2h(\ — sin.  i),^     ,,        ,    m  •           ^„^  2h 

=—r ^-z^(Bt.  Young's  Trig,  art  26)  = ~^- 

To  determine  the  greatest  height  M'P  above  AB'  (fig.  97),  we 
must  make  PM  —  MM'  a  maximum,  or,  since  MM'=a;tan.  i, 

M'P=tan.  e.  X  —  —5 • tan.  ia?=max. 

4  h  cos.*^  Q 

X 

.'.  tan.  0 —  — T tan.  2=0 

2  h  cos.39 

_,         „    .  "    -N      2  A  cos.  5  sin.  (5  —  i) 

.'.  x=2  h  cos.2  9  (tan.  e  —  tan.  i)  = 7-^^ . 

cos,  I 

This  value  of  x  being  substituted  in  the  above  expression  for  M'P, 

,  .  ,       .  .       sin.  (e  — i)     .     , 

which,  since  tan.  e  — tan.z= ^^ -,  is  the  same  as 

cos. 9  cos.  I 

X       sin.  {9  —  I)  X 

cos. 9         COS. %  4  h  COS.  9 

gives  for  M'P,  when  a  maximum,  the  value 

-^     _2  A  sin.  (9  —  i)  sin.  {9  — i)       sin.  {9  —  *')>_'*  ^i"-^  (^ — **) 

COS.  i  *       COS.  i  2cos.i  cos.^  £ 

(114.)  Collecting  together  the  principal  results  of  the  preceding 

propositions,  we  have  the  following  formulas  : 

I.  When  the  Plane  is  horizontal. 


f9  h 

Range=2  A  sin.  2  9,  time=2  sin.e-^ — 

Greatest  range  =2  h 
Greatest  height  =h  sin.^  9. 

II.  When  the  Plane  is  oblique. 

—             ^  -COS.  esin.  (9  — i) 
Range =4  h ^ 

^  COS.^  t 

COS.  t  ^     £• 


136  ELEMENTS    OF    DYNAMICS. 

2/j 


Greatest  ranffe=- 


l-|-sin.  i 

^  ...       hsu\*(d—  i) 

Greatest  heiffht= ^--; — -  . 

^  cos.'i 

These  equations  contain  the  whole  tlieory  of  projectiles  in  vacuo ; 
tliey  may  all  be  deduced,  independently  of  analysis,  by  the  aid  of 
common  geometry,  and  a  few  well  known  properties  of  the  para- 
bola. See  the  second  volume  of  Br.  Huttoii's  Course  of  Mathe- 
matics. 


CHAPTER  III. 

ON  THE  RECTILINEAR  MOTION,  PRODUCED  BV  A  VARIABLE  FORCE. 

(115.)  We  shall  now  proceed  to  show  the  application  of  the 
general  formulas,  at  art.  (106),  to  cases  of  rectilinear  motion,  pro- 
duced by  forces  varying  in  intensity  according  to  some  known  law. 
This  variation  is  generally  according  to  some  function  of  the  dis- 
tance of  the  moving  body  from  the  fixed  point,  which  is  regarded  as 
the  centre  of  force,  although,  in  some  cases  which  nature  presents, 
the  variation  is  also  dependent  upon  other  circumstances ;  as,  for 
instance,  when  the  motion  takes  place,  not  in  free  space,  but  in  a 
resisting  medium,  where,  it  is  obvious,  the  body  will  be  hindered 
from  obeying  the  full  influence  of  the  attracting  force  hj-^  a  resisting 
force,  varying  in  some  manner  with  the  velocity.  These  particu- 
lars will  be  considered  in  prob.  III. 

Problem  I. — (116.)  To  determine  the  vertical  motion  of  a  heavy 
body  towards  the  earth  ;  the  force  of  gravity  varying  inversely  as 
the  square  of  the  distance  from  the  centre. 

Call  the  radius  of  the  earth  r,  the  distance  of  the  body  from  the 
centre  at  the  commencement  of  motion  a,  and  the  distance  at  any 
time  r',  after  the  commencement  a; ;  then  by  the  hypothesis  the 
intensity  of  the  force  F,  at  the  time  t",  will  be  given  by  the  propor- 
tion 

—  •  —  ::<>•:  F  .".  F= —  i?"= .* 

Having  got  an  expression  for  the  force,  the  next  object  is  to  de- 
duce that  for  the  velocity.     Referring  to  equation  (D),  we  have 

|.2  p.  Ay.      2  r^  £* 

J = — |-C ;  the  constant  C  will  depend  upon 

•  See  note  in  next  page. 


ox    RECTILINEAR    MOTION.  137 

the  initial  velocity  of  the  body,  that  is,  upon  the  velocity  which  it 
has  at  the  distance  a,  where  gravity  begins  to  act ;  if  this  velocity 

is  0,  then 2__|_C==0  .-.  C= 2_  ;  and  thus  the  velocity 

a  a  ^ 

of  the  body  has  at  any  time  t",  that  is,  after  having  fallen  from  the 

distance  a  to  the  distance  x  is   completely  determined,  it  is  v^=: 

2r^  g     2r^  g     2r^  g(a  —  x)  .1     ,    j 

—= ^-^ ;  and  when  the  body  arrives  at  the 

X  a  ax  ■' 

surface  of  the  earth,  that  is,  when  x-=r,  it  will  have  acquired  a 
velocity  expressed  by 

,1rg{a  —  r)        ,  •  ,     .^      ■    ■  a  ■.      ■      « —  '^ 
v=^ — 2-):^ ;  which,  II  a  is  infinite,  since 


is  then  1,  becomes  ?;=v'  2  rg;  so  that  the  velocity  can  never  be  so 
great  as  this,  however  far  the  body  may  fall,  and,  hence  if  it  were 
possible  to  project  a  body  vertically  upwards  with  this  velocity,  it 
would  go  on  to  infinity  and  never  stop.  Of  course  this  is  on  the 
supposition  that  there  is  no  resisting  medium  nor  other  disturbing 
force.  Taking  the  radius  of  the  earth  at  3965  miles,  the  last  ex- 
pression for  V  will  be  t)  =  6*9506  miles  ;  so  that  if  a  body  were  to 
he  projected  upwards,  with  a  velocity  of  about  seven  miles  a  second, 
and  were  to  experience  no  resistance,  it  would  never  return  to  the 
earth. 

It  remains  now  to  determine  the  time  t" ;  and  for  this  purpose  we 

have  the  equation -jj-z=v=.r\  _ ;  from  which  we  get 


dt  ax 

s/a 


rv/2o-\   a  —  X 


■dx 


t=:r^^^f^=~-dx; 


r-s/l  g  y/\ax — x'^\ 
s/a       p    —  X 

'Tx/'ig  J  Vax  —  x^ 
this  integral  may  be  immediately  found  by  means  of  the  general  ex- 
— 

*  In  the  expression  —  for  the  velocity  at  (p.  123),  s  is  the  space  passed 

d  (a  —  x)  dx 

dt        ~        dt  '■ 
dx  dv  </2  X 

case,  V  = —  — ,  and  —  =  F  ^  — "Ti"'  ^^^'  generally,  whenever  F  tends  to  di- 

..-.,.  ds 

minish  the  space  s,  the  increment  A  s  being  always  negative,  the  expression   — , 

d^s 
for  the  velocity,  as  also  the  expression  -j-r-  for  the  force  must,  obviously,  be  like- 

dt^ 

wise  negative. 

h2  18 


138  ELEMENTS    OF    DYNAMICS. 

pression  at  the  top  of  page  46  in  the  Integral  Calculus,  or  we  may 

proceed  thus :  to  the  numerator  of  the  expression  to  be  integrated 

add  2  adx,  and  then  subtract  the  same  quantity  from  it,  and  we 

shall   thus   convert  the  differential   into  two   others,  of  which  one 

will  be  immediately  integrable  by  the  rule  for  powers ;  thus  we 

iadx — xdx  ^  adx 

shall  have  the  two  expressions  — — — ; 

^  x/\ax—x-'\  y/{ax—x''\ 

the  integral  of  the  first  of  these  is  obviously  ^\ax  —  x"]  ;  that  of 

the  second  is  the  elementary  integral,  5  a  coversin."^  —  x  (Int. 

%*' I  i/iiA^  ii       '  2x a 

J  Calc.  p.  10),  or,  which  is  the  same  thing,  5  a  cos.-^  ( ). 

Consequently 

t=       ,^  x/ox  — a^^  +  l  acos. -1  ( )     ; 

which  expresses  the  number  of  seconds  elapsed  in  moving  from  the 
distance  a  to  the  distance  x,  from  the  centre  of  attraction.  This 
expression  needs  no  correction,  because  a  —  x,  and  t  become  0  at 
the  same  time.  When  x — r,  that  is,  when  the  body  arrives  at  the 
surface  of  the  earth,  the  number  of  seconds  elapsed  will  be 

^=-^--    ^/ar  — r''  +^fl  cos.  "»( )   ; 

which  is  evidently  infinite,  when  a  is,  although  the  velocity,  as  we 
have  before  seen,  is  finite. 

If  the  attracting  body  be  considered  as  merely  a  point,  then  x  may 
at  length  become  0,  and  the  whole  time  of  falling  to  the  centre  from 
the  distance  a  will  be  expressed  by 

t= — — — xiacos.   1  —  1=- 717— a-. 

r^2g  2rV2^ 

If  the  body  fall  from  any  other  distance  a'  the  expression  for  t  would 
be  similar,  so  that  t^  :  t'^  :  :  a^  :  a'^,  that  is,  the  squares  of  the  times 
of  falling  from  rest  to  the  centre  offeree  are  as  the  cubes  of  the 
distances  from  which  they  fall. 

Problem  II. — (117.)  To  determine  the  motion  of  a  body  at- 
tracted towards  a  fixed  centre,  the  force  varying  directly  as  the  dis- 
tance. 

This  is  the  law  of  force  which  would  attract  a  material  point  at 
liberty  to  move  along  a  perforation  from  the  surface  to  the  centre  of 
the  earth.  For  the  universal  law  of  nature  being  this,  that  every  par- 
ticle of  matter  attracts  with  a  force  varying  in  intensity  inversely  as 
the  square  of  the  distance  at  which  it  acts,  it  follows  that  the  at- 
tractive forces  of  homogeneous  spheres  must  vary  directly  as  their 


ON    RECTILINEAR   MOTION.  139 

masses,  or  as  the  cubes  of  their  radii,  and  inversely  as  the  squares 
of  the  distances  of  their  centres  from  the  attracted  point.  Now  it  is 
shown,  by  writers  on  Physical  Astronomy,  that  the  attraction  of  a 
sphere  is  the  same  as  if  its  entire  mass  were  concentrated  in  its  cen- 
tre ;  it  is  shown,  moreover,  that  a  particle  placed  any  where  within 
a  spherical  shell  will  remain  at  rest,  being  equally  attracted  in  all 
directions.  If,  therefore,  a  particle  be  placed  below  the  surface  of 
the  earth,  and  at  the  distance  x'  from  the  centre,  it  will  be  moved 
only  by  the  force  which  resides  in  the  inner  sphere  of  radius  x,  as 
it  is  kept  at  rest  by  the  influence  of  the  shell,  whose  thickness  is 
r — x;  hence,  from  what  has  just  been  said, 

which  shows  that  the  force  varies  directly  as  the  distance  x.  Having 
thus  got  the  value  of  F  we  have,  as  in  last  problem, 

^t;2=   f 5-xdx^—-^X^  +  C. 

«/  r  2  r 

If  the  body  be  merely  dropped  into  the  hole,  v  will  be  0,  when  x=7-, 

.-.  0=  —  ^r^+2  C  .-.  C  =  h  gr  .:  V  =J^{r^—x^)  -  (1); 

which  is  the  velocity  of  the  body  at  any  distance  x  from  the  centre 
when  the  body  reaches  the  centre,  that  is,  when  a:=0,  the  velocity 
isv=^\gr\'---{2).  This  velocity  must  be  spent  before  the  body 
will  stop,  and  as  the  motion  after  passing  the  centre  will  be  retarded 
according  to  the  same  law,  as  it  was  before  accelerated,  the  body 
will  continue  to  move  till  it  reaches  the  opposite  point  of  the  earth's 
surface,  where  it  would  stop ;  but  being  again  attracted  by  the 
same  force  as  at  first,  it  will  return  and  pass  through  the  centre  to 
the  point  of  departure,  and  will  thus  move  backwards  and  forwards 
continually. 

To  determine  the  time  from  the  departure  of  the  body  till  its  arri- 
val at  the  distance  x  from  the  centre,  we  have,  as  usual, 


-^=''=J7('"---)  ■••*=■ 


r  dx 

•V  — - 


g    V\t'—x^] 
,r  p       dx  T  X 

S         J  ^\r^—x^\  g  r 

which  requires  no  correction,  since  ^=0,  when  x=r.    For  the  time 
of  reaching  the  centre  we  have,  by  making  a?=0, 

By  making  x=-  —  r  we  have,  for  the  time  of  passing  through  the 


140  ELEMENTS  OF  DYNAMICS. 

whole  diameter  t-=Tty/  —  ••••  (4) ;  which  is  twice  the  time  of  fall- 

ing  to  the  centre  as  it  ousjht  to  be. 

If  the  body  do  not  bepin  to  move  from  the  surface  of  the  earth, 
but  from  some  point  within  it  at  the  distance  r',  instead  of  r,  from 

the  centre  then  — r',  will  be  the  force  at  the  commencement  of  mo- 
r 

tion  instead  of  g-,  and,  therefore,  substituting  this  for  g,  and  r'  for  r, 
the  expression  for  t  will  be  ;=v'— xcos.  -*—  ;  and,  consequently, 
whena:=r',  we  have  for  the  time  of  reaching  the  centre 

the  same  as  the  expression  (3).  Hence,  at  whatever  point  within 
the  surface  of  the  earth  the  body  be  placed,  it  will  reach  the  centre 
in  the  same  time. 

In  order  to  find  this  time,  take  the  radius  r,  of  the  earth,  equal 
to  3965  miles,  and  we  shall  have  r '=21' 7"i,  which  will  be  the 
time  occupied  in  passing  to  the  centre,  however  near  to  it  the  body 
be  placed. 

It  is  obvious  that  if  g  represented  the  energy  of  any  other  force, 
instead  of  gravity,  at  the  distance  r  from  the  centre,  the  reasoning 
and  conclusions  would  be  the  very  same,  so  that  when  a  body  is 
attracted  from  a  state  of  quiescence  by  any  centre  of  force,  varying 
in  intensity  directly  as  the  distance,  the  whole  time  of  passing  to' 
the  centre  will  be  the  same  from  whatever  point  the  motion  com- 
mences, whether  from  a  point  infinitely  distant,  or  from  a  point  in- 
finitely near.  Hence  the  body  would  pass  over  an  infinite  space  in 
a  finite  portion  of  time,  but  then,  by  hypothesis,  the  force  at  the 
commencement  of  motion  must  be  infinitely  great. 

We  shall  now  consider  the  motion  of  a  body  near  llie  surface  of 
the  earth,  taking  into  account  the  resistance  of  the  air,  which  we 
have  hitlierto  neglected. 

Problem  HI. — (118-)  To  determine  the  vertical  motion  of  a 
heavy  body  near  the  earth's  surface,  considering  the  resistance  of 
the  air  to  vary  as  the  square  of  the  velocity. 

If  we  represent  the  resisting  force  at  any  time  t'\  after  the  com- 
mencement of  motion  by  f,  and  the  velocity  generated  by  u,  then, 
by  hypothesis  f=niv^,  m  being  constant  for  all  velocities,  and 
which  can  be  determined  only  from  experiment.  Hence  the  force 
F,  accelerating  the  body,  is  F:=^  —  inv^  ;  so  that  here  we  have  F 
as  a  function  of  the  velocity,  and  not  of  the  space  as  heretofore ; 
therefore,  since 


ON    RECTILINEAR    MOTION.  141 

.^.  ^      dv  dv  ^  dv 

(D)F=-y-,  we  have  -      =g  —  mv^  .-,  dt=- 


dt  dt  g  —  mv^ 

The  second  meniber  of  this  equation  may  be  integrated  by  the  me- 
t!iod  of  rational  fi&tions  ;  or  we  immediately  see  that  the  expression 

•     ,  /        ^^^  ,  dv        \  I 

IS  the  same  as  (  — — - — -^ 1 ; ;—  )  x  — — i-;    consequently, 

\  gi-\-mi  V     gi — imv/      2  g-3 

/    P dv n dv        \         1 

\J     gk-\-mh  V       *^   gh  —  mk  v)      1  g\ 

1  -^  "'^'-     '■ 

{log.  (^+m3  u)  —  log.  (^  —  w^v)|+C 


1m\  gh 


=  '°g-^^r  ••••(!); 


the  constant  being  0,  because  the  time  and  the  velocity  begin  to- 
gether. 

This  determines  the  time  of  the  motion  necessary  to  generate  a 
given  velocity.  To  find  the  velocity  when  the  time  is  given  it  will 
be  necessary  to  disengage  the  equation  from  logarithms,  putting, 
therefore,  e  for  the  hyperbolic  base,  we  have,  since  log.  e=l, 

o.    J, ,  1       gi+mhv         it\/ mg      gh+mhv        .. 

'It  y/  nig    log.  e=log.  ^ i —  .-.  e  —-^ , —  .  (2) : 

from  which  v  may  be  found  when  t  is  known. 

Since  e  exceeds  unity,  the  first  member  of  this  equation  increases 
with  t,  and  is  infinite  when  t  is,  consequently,  as  t  approaches  to 
infinity  the  denominator  gi —  mkv,  must  approach  to  0,  but  when 

a- 

it  is  actually  0,u= v'  --  •  (3);  so  that  the  longer  the  body  is  in  mo- 
tion, that  is,  the  greater  the  space  through  which  the  body  moves 
in  a  medium,  varying  in  resistance  as  the  square  of  the  velocity, 
and,  towards  an  attractive  force,  varying  in\-€rsely  as  the  square  of 
the  distance,  the  nearer  will  the  velocity  approach  to  constancy. 

Having  found  the  velocity  we  may  readily  determine  the  space  s, 
through  which  the  body  has  passed  to  acquire  it.     For,  by  (D), 
vdv=Fds=-{g  —  mv^)  ds. 

vdv  1   ,        .  V     ^ 

-log.  {g  —  'mv^)-\-C. 


'=/- 


g< — mv^  2m 

To  determine  C  put  s=0,  then  v=0, 

.'.  0= log.  2-+C  .-.  C=- —  log.  g 

1  mv^ 

.'.  s= log.  (1 ). 

2m      ^  ^  g  ^ 


142  ELEMENTS  OF  DYNAMICS. 

If  the  space  be  already  known  the  acquired  velocity  may  be  found 
by  this  equation. 

If  the  body  is  projected  with  a  velocity  v  ,  in  the  resisting  medium 
in  opposition  to  the  force,  then  the  motion  becomes  retarded,  both 
gravity  and  the  force  ^nr"  conspiring  to  stop  the  body ;  hence  the 
tttardive  force  is  F= — g  —  mv^ 

dv                      .             ,,  dv 

dt         °  g+mv" 

V  gm  ^  s 

We  may  determine  C  from  tlie  circumstance  that  at  the  com- 
iiiencement  of  motion,  that  is,  when  <=0,  v^=v^,  so  that 

0=--L^ftan.-J-^i,,|+C; 
V  gm  ^  S 

therefore,  subtracting  the  former  equation,  from  this,  we  have 


t=- 


0^m  ^  g  ^  g 


^/  gm  yg  \g 

from  which  equation  v  may  be  determined  for  any  proposed  time 
t".  It  remains  now  to  deduce  the  expression  for  the  corresponding 
space  ;  for  this  purpose  we  must  employ  the  formula  (E),  which 
gives 

/v    ,                    /»   vdv 
-=-  dv  .'.  — s=  I ; 

that  is, — s=- — log.  (fi'+mv*)  +  C.     As  s=0,   whenu=y,,  we 
2m 

have  0=  — log.  {g^mv,^)+C 

1  Iff  -l-wio  ' 

.-.  s= llog.  (g-\-mv, ")  — log.  (g+mv')=- — log.  \^—^ — ^L 

2m  '    ^  ^°  ^  '         ^  ^^  '1m     ^    ^g  frnv""  * 

When  the  retardive  force  has  destroyed  the  motion,  that  is,  when 

1  7HV    ^ 

v=0  we  shall  have  s=- —  log.  \l-\ —}  ;  which  expresses  the 

height  to  which  the  body  will  reach  when  projected  with  the  velo- 
city V.  It  will  not  acquire  so  great  a  velocity  in  returning  to  the 
earth,  because  the  accelerating  force  is  only  g  —  mv^. 


THE    MOTION    OF    A    FREE    POINT.  143 

SECTION  II. 
ON  THE  THEORY  OF  CURVILINEAR  MOTION. 

(119.)  We  now  come  to  discuss  the  general  theory  of  the  mo- 
tion of  2.  free  point,  or  material  particle,  independently  of  any  re- 
striction as  to  the  nature  of  the  path  it  is  compelled  to  take. 

We  say  point  instead  of  body,  because  we  do  not  propose  to 
take  into  consideration,  in  the  present  section,  any  circumstances 
of  the  motion  which  may  be  dependent  upon  the  mass  of  the 
moving  body.  Whenever,  therefore,  in  the  course  of  this  section, 
we  speak  of  the  motion  of  a  body,  it  must  be  noticed  that  we  con- 
sider the  acting  forces  to  apply  themselves  equally  to  all  the  par- 
ticles of  the  body,  and  that  these  particles  exert  no  power  them- 
selves sufficient  to  modify  the  motion.  This,  indeed,  is  the 
hypothesis  upon  which  the  investigations  in  the  preceding  section 
are  founded ;  where  we  have  considered  the  motions  of  bodies 
chiefly  in  reference  to  the  force  of  gravity. 

But,  in  fact,  as  already  hinted  at  (118),  all  bodies  in  nature  exert 
a  mutual  influence  on  each  other,  and  the  intensity  of  this  in- 
fluence varies  with  the  mass  from  which  it  emanates.  Two  bodies 
then,  M  and  m,  at  liberty  to  obey  their  mutual  attractions,  ap- 
proach each  other  in  virtue  of  the  force  resident  in  M,  combined 
with  the  force  resident  in  m ;  and,  therefore,  the  distance  of  their 
centres  at  any  time  will  be  the  same  as  if  the  sum  of  the  attractions 
due  to  M  and  m  were  combined  in  M,  and,  instead  of  the  other 
body  m,  a  single  particle  were  placed  at  its  centre ;  so  that  we 
should  thus  have  to  consider  only  the  motion  of  a  single  particle  or 
free  point  acted  on  by  a  single  centre  of  force,  viz.  the  centre  of 
M's  attraction.  We  may,  therefore,  after  having  attributed  a  proper 
value  to  the  attractive  force  tending  to  M's  centre,  disregard  the 
mass  of  the  moving  body  m ;  and  when,  in  the  course  of  the  present 
section,  we  speak  of  the  motion  of  a  body  about  a  fixed  centre  of 
force,  we  consider  the  influence  of  the  moving  body  to  have  been 
transferred  to  that  centre  as  above,  and,  therefore,  the  body  to  move 
as  a  free  point. 

Let  us  now  proceed  to  investigate  the  general  equations  of  mo- 
tion. 


144  ELEMENTS  OF  DVNAMlCS. 


CHAPTER  I. 


ON    THE    GENERAL   EQUATIONS    OF    THE    MOTION    OF    A    FREE    POINT. 

(120.)  When  a  material  particle  moves  in  a  curve  line,  it  is  ob- 
vious that  its  direction  at  any  point  of  its  path  is  in  the  tangent  at 
that  point,  alonff  which,  if  it  were  there  left  to  itself,  it  would  pro- 
ceed with  a  uniform  velocity  :  and  its  curvilinear  course  is  kept  up 
only  by  the  continual  influence  of  some  force  or  forces  which  at 
every  instant  deflect  it  from  the  rectilinear  course  it  tends  to  pur- 
sue, in  virtue  of  its  inertia. 

As  far  as  efiects  are  concerned  we  may  consider  a  body  thus 
moving  to  be  impelled  along  the  curve  by  an  accompanying  force, 
varying  in  intensity  conformably  to  the  circumstances  of  the  mo- 
tion, and  we  know  that  the  value  of  this  force  at  any  time  t", 
during  which  the  arc  s  has  been  described,  will  be  expressed  by 

~dF' 
If  this  same  force  had  a  statical  effect,  or  which  is  the  same 
thing,  if  it  were  directly  opposed  by  an  equal  force  F,  we  might 
then,  instead  of  the  first  force,  substitute  three  others  acting  on  the 
point  and  in  the  directions  of  three  rectangular  axes ;  if  o,  (3,  y,  be 
the  angles  which  these  form,  with  the  line  along  which  the  point 
tends  to  move,  then  the  values  of  these  three  forces  will  be 

F  cos.  a,  F  cos.  ^,  F  cos.  y ; 
so  that  tlie  point  will  have  the  same  tendency  to  move  under  the 
influences  of  these  three  forces  as  under  the  influence  of  the  ori- 
ginal force  F ;  these,  therefore,  are  fitted  to  produce  the  same  ac- 

celeration  -r—  as  F  ;  let  us  see  what  acceleration  each  alone  is  fitted 
or 

to  produce.     In  order  to  this  let  us  first  remark  that  since 

dx  (is  dx     d^  s  d^  x    ,  . 

-—  =C0S.  a,  -pCOS.  a  =-p.'.— 7— COS.a  =— :—  .  (1); 

ds  dt  dt       df'  dt""     ^  ' 

then,  by  the  principle  stated  at  p.  121, 

-,    ^  d^s    d^s  d^x         ,„, 

r  :  t  COS.  o  :  :  —r—  :  -;— cos.  o=  — —  ....  (2) ; 

d^  X 

hence  the  acceleration  which  F  cos.  a  is  fitted  to  produce,  is  -^-^, 

and,  in  like  manner,  the  accelerations  due  to  F  cos.  /3  and  F  cos.  y 

rf^V        .d'z 
are  —r— ,  and  -r-^ ;  x,  y,  and  z  being  the  co-ordinates  of  the  point 


THE    MOTION    OF    A    FREE    POINT.  145 

at  the  instant  t".  Thus  tlien  the  consideration  of  the  curvilinear 
motion  of  a  material  point  in  space  is  reduced  to  the  consideration 
of  the  rectilinear  motions  of  its  three  projections  along  three  rec- 
tangular axes,  and  which  describe  the  rectilinear  spaces  x,  y,  s, 
Avhile  the  body  itself  describes  the  curve  s.  Calling  the  forces 
along  these  axes  X,  Y,  and  Z,  we  have 

and  these  are  the  general  equations  of  the  motion  of  a  free  point. 

mi         T     •  •        ,.1  •      •  dx  dy        ,dz        ,    , 

Ihe  velocities  of  the  proiections  are-r-,  -r^,  and^-,  and  the  ve- 
^    "^  dt    dt  dt 

ds 
locity  of  the  point  itself  is-^,  but  (Diff.  Calc.  p,  215,) 

rfs"  _  dx"" +dy''-i-  dz^  ,  ,  ^ 

IF  ~  df^  ""^  ^ ' 

from  which  expression  it  follows  that  if  the  lines  which  represent 

the  velocities -7- ,  -^,  -r-,  be  taken  for  the  edges  of  a  rectangular 
dt  dt   dt  ^  ^ 

ds 
parallelopiped,  the  diagonal  of  it  will  represent  the  velocity— of 

the  point  in  space ;  and  it  equally  follows  from  the  equation  (2), 

combined  with  the  two  similar  equations  furnished  by  the  other 

projections,  that  if  the  lines  which  represent  the  accelerative  forces 

d^  X  d^  y  d^  z 

—fji  -yf  5  -T-j,  be  taken  for  the  edges,  the  diagonal  will  represent 

d^  s         . 
the  accelerative  force  y^,  which  acts  upon  the  point  in  space. 

It  thus  appears  that  the  velocity  which  a  body  actually  has  may 
always  be  decomposed  into  three,  directed  according  to  three  rec- 
tangular axes ;  and  the  accelerative  force,  by  which  a  body  is  ac- 
tually influenced,  may  always  be  decomposed  into  three,  directed 
according  to  three  rectangular  axes  ;  and  conversely  such  a  system 
of  velocities,  or  of  forces,  may  be  always  compounded  into  one. 
When  the  motion  is  in  a  plane  curve,  one  of  the  components  is  of 
course  0. 

If  we  differentiate  the  equation  (4),  relatively  to  the  independent 
variable  t,  we  shall  have 

ds^        d^'x.  dx-\-d^y  .  dy-^d'^z.  dz 

"^IF-^  dP ' 

but,  in  virtue  of  equations  (3),  the  second  member  of  this  equation  is 

2{^dx+Ydy+Zdz); 
consequently,  returning  to  the  integral,  we  have 
N  19 


146  ELEMENTS  OF  DYNAMICS. 

(1) .  .  .  .  ^=,«=2/(X  (/^  +  Y  ch,  +  Z  dz)  ....  (2). 

We  infer,  therefore,  that  when  the  component  forces  X,  Y,  Z,  are 
known  functions  of  the  co-ordinates  .r,  y,  z,  for  every  point  of  the 
body's  path,  or  trajectory,  as  it  is  called,  and  when,  moreover, 
these  functions  are  such  as  to  render  \dx-\-\dy-\-'Aiiz  an  exact 
differential,*  then  the  velocity  of  the  moving  body  will  be  deter- 
minable from  this  equation.  The  complete  integral  of  (2)  will  in- 
volve an  arbitrary  constant,  which  can  only  be  determined  from 
previously  knowing  the  velocity  at  some  known  point  of  the  tra- 
jectory, which  is  the  same  as  if  we  knew  the  point  at  which  the 
motion  commenced. 

Suppose  the  general  integral  of  (2)  were 
v'^f{x,y,z,)  +  C; 
if  we  knew  that  the  velocity,  at  the  point  (a,  b,  c,),  were  t\  wc 
should  have  v^"=f(a,  b,  c,)-j-C 

.:v'-v,'^=f(x,y,z)-f{a,b,c). 
From  this  equation  it  appears  that  when  we  know  the  functions  that 
X,  Y,  Z,  are  of  x,  y,  z,  and  the  velocity  at  any  point  (a,  b,  c),  we 
may  find  tlie  velocity  at  any  other  point  [x,  y,  z,)  merely  from 
knowing  its  co-ordinates,  without  requiring  to  know  either  the  form 
of  the  curve  between  the  two  points  (a,  b,  c),  (.r,  y,  r),  or  the  time 
of  describing  it ;  in  bodies,  therefore,  which  move  in  curves,  re- 
turning into  themselves,  the  velocity  is  always  the  same  at  the  same 
point. 

(121.)  It  is  an  important  fact  that  tlie  differential  in  equation  (2) 
is  always  exact  whenever  the  body  moves  under  the  influence  of  a 
force  emanating  from  a  fixed  centre,  or,  indeed,  when  any  number 
of  fixed  centres  act  on  the  body,  provided  always  that  the  intensity 
of  each  force  is  a  function  of  the  distance  of  the  point,  or  body,  on 
which  it  acts. 

Let  there  be  but  one  such  centre,  then  placing  the  origin  of  the 
co-ordinates  there,  and  calling  r  the  distance  of  the  moving  body 
from  it,  at  any  point  of  its  path,  we  may  express  the  force  acting 
on  it  by/r,  the  form  of  the  function  /  being  known  ;  and,  by  re- 
solving this  force  according  to  the  three  axes,  we  have  the  compo- 
nents 

X=./r.f  Y=/r.f  Z=/r.f 

Substituting  these  expressions  in  the  general  equation  (2)  we  have 

*  In  order  to  this  these  functions  must  satisfy  the  conditions 
dX     dY   (IX     iJZ   dY      dZ 

-^=^'-^=  di'  17=^'  ^^''  ^«'-  ^'^^  P-  '"^-^ 


THE    MOTION    OF    A    FREE    POINT.  147 

r2==2//r  . Z^L_^_Z ;    but  since  r^=::X^+y^+z^  .-.  rdr 

=  x  dx-{-y  dy-\-z  dz,  therefore,  by  substitution,  ij^  =  2/"/)' .  dr 
....  (1) ;  so  that  the  velocity  is  always  determinable. 

The  same  may  be  shown  generally  as  follows  : 

Having  chosen  the  axes  of  reference,  let  the  co-ordinates  of  one 

of  the  fixed  centres  be  a,  b,  c,  and  let  R  be  the  force  exerted  by  this 

centre  on  a  point  at  the  distance  r  from  it.     Now  the  cosines  of  the 

angles  which  r  makes  with  the  axes  are,  severally, 

X  —  ci  xi  "—  h   z  '•"—*  c 

, , :  so  that  the  three  components  of  R  are 

r  r  r 

x— a  ^y  —  b      z—c 
r  r  r 

In  like  manner,  for  the  components  of  R^,  R^,  &;c.,  the  forces  simul- 
taneously acting  on  the  point  from  a  second,  a  third  centre,  (fee, 

we  have  R, -,    U,- -,    R, 

T  r  r 

nf-4^,  Ry--=^,  E.17^ 

r„  r„  Ta 


R,£=±.,    R„^_=^,    R.i=:fa; 

n  n  n 

so  that  if  we  add  together  each  of  these  three  vertical  columns  of 
components,  we  shall  have  the  expressions  for  X,  Y,  and  Z,  to  be 
introduced  into  the  general  formula  (2).  But,  before  performing 
this  addition,  we  should  remark,  that  since 

...  r„  rfr„=(.r— a„)  dx-\-(y—b^)  dy  +  {z  —  cj  dz; 
consequently,  if  we  divide  this  equation  by  i\,  and  multiply  by  R„, 
and  then  put  n  successively  equal  to  0,  1,  2,  &c.,  we  shall  have 

Rrfr=R f/.r+R  ^ dy+- dz 

r  r       ^         r 


R,  (7r,=R, ^  dx+Yi^  y- — ^  dy+- — ^  dz 


R„  dr=n,,'^~^dx+^y—^dy+^—-^  dz 


.-.  R(/r+R,rfr,...  +  R„rfr„=       Xdx      +       Y  dy      +Zdz; 
hence,  if  R„  be  a  function  of  r„  whatever  be  n,  the  first  member  of 
this  equation  will  be  always  integrable,  and  Ave  shall  have 
v'^=2fIldr+2fR^dr +2/R„rfr„ 


148  ELEMENTS  OF  DYNAMICS. 

(122.)  We  shall  notice  in  this  place  a  remarkable  property  con- 
nected with  the  motion  of  a  body  about  a  single  centre  of  force,  viz. 
that  tlie  trilineal  spaces  described  by  the  radius  vector  or  line  joining 
the  moving  point  and  fixed  centre  are  to  each  other  as  the  times  of 
describing  them,  and  this  whatever  be  the  law  according  to  which 
the  intensity  of  the  attractive  force  varies. 

When  a  body  acted  upon  by  a  single  centre  of  force  moves  in  a 
curve  line,  we  may  consider  such  motion  to  arise  from  a  primitive 
impulsion  given  to  the  body  which  would  alone  have  caused  it  to  de- 
scribe a  straiglit  line,  but  being  continually  acted  upon  by  a  force  out 
of  this  line  it  is  deflected  from  this  path  at  tlie  very  commencement 
of  motion,  leaving  it  a  tangent  to  the  path  it  actually  takes  at  the 
point  of  projection ;  as  moreover  nothing  draws  the  body  out  of 
the  plane  in  which  the  centre  of  force  and  line  of  projection  are 
situated,  the  path  of  the  body  must  be  a  plain  curve.  Hence,  placing 
the  origin  of  the  rectangular  axes  at  the  centre  of  force  S  (fig.  98), 
and  taking  P  for  the  place  of  the  body  at  any  time  t",  we  have 

X=  —  F  cos.  a,  Y=  —  F  cos.  p 

r  r 

the  negative  signs  being  used  because  F  tends  to  diminish  the  spaces 
X  and  y ;  consequently  the  equations  of  motion  are 
d^  X  X    d"  y  y 

To  eliminate  F  multiply  the  first  of  these  equations  by  y  and  the 

second  by  x,  and  subtract  the  products  ;  there  results 

yd'  X  -  xd'  y_^ 

TlF         ~"  •  •  •  •  ^^' 

an  expression   altogether  independent  of  the  value  of  F,  so  that 

whatever  results  from  it  will  hold  even  when  F  is  repulsive. 

Now  it  is  easy  to  perceive  that  the  numerator  of  this  expression 

is  the  differential  of  ydx — xdy,  therefore,  multiplying  by  dt  and 

iidx xdu 

integrating,  we  have  - — -—^ —=  C  ....  (2) 

.-.  fydx — f  xdy=:Ct-\-Q^, 
or2fdyx  —  xy=Ct  +  CK  .  .  .  (3). 
Let  us  inquire  into  the  geometrical  signification  of  the  first  member 
of  this  equation.     The  term  2  fydx  obviously  expresses  twice  the 
area  SP'PM,  and  the  term  xy  is  twice  the  triangle  PSM,  conse- 
quently the  equation  (3)  is  the  same  as 

2  sector  SP'P=C/+C,. 
Suppose  t"  to  commence  when  the  body  is  at  P',  then  since  when 
/— -0,  the  sector=0  .'.  C,  =0,  consequently 


THE    MOTION    OF    A    FREE    POINT.  149 

c,^  r^.     ^.       sector  SPP'      ,_ 
2  sector  SP  F'=Ct .-. ==|  C, 

that  is,  as  announced  above,  the  sectors  described  are  proportional 
to  the  times  of  describing  them,  and  therefore  equal  areas  are  de- 
scribed about  S  in  equal  times:  this  remarkable  and  important 
property  is  called  the  general  principle  of  equal  areas. 

(123.)  In  order  to  complete  the  theory  of  curvilinear  motion,  it 
may  be  as  well  to  repeat  here  the  general  expression  given  at  (103) 
for  the  force  fitted  to  produce  the  motion  which  a  body  actually  has 
at  any  instant  if  immediately  urged  by  that  force  in  the  direction  of 
its  motion,  that  is,  in  the  direction  of  a  tangent  to  the  curve  at  the 
point  where  it  is  at  that  instant.     Calling  such  a  force  S,  its  value  is 

S  =  ^alsoi  w''=/Srf5. 

The  force  S  may  be  called  the  tangential  force ;  we  see  from  the 
second  equation  that  upon  this  force  the  velocity  of  the  body  in  its 
path  wholly  depends,  and  it  is  wholly  expended  in  producing  this 
velocity  ;  if,  therefore,  all  the  forces  which  influence  the  body  at  any 
particular  point  were  decomposed,  each  into  two,  one  in  the  direc- 
tion of  the  tangent,  and  the  other  in  the  direction  of  the  normal,  the 
sum  of  the  former  components,  that  is,  the  tangential  force,  would 
determine  the  velocity  and  direction  of  the  body's  motion  at  that 
point.  It  follows,  therefore,  that  if  among  the  forces  which  act 
upon  a  body  there  be  any  which  always  act  in  the  direction  of  a 
normal  to  its  path,  the  components  of  these  must  necessarily  destroy 
each  other  in  the  expression  (2),  (p.  146,)  for  the  velocity  along  the 
curve ;  because,  as  just  observed,  this  velocity  is  wholly  due  to  the 
tangential  force. 

(124.)  Before  closing  this  preliminary  chapter,  we  shall  briefly 
show  how  the  equations  of  motion  investigated  in  the  preceding 
section,  are  to  be  deduced  from  the  more  general  theory  laid  down 
in  the  present  chapter. 

As  a  first  application,  let  it  be  required  to  determine  the  motion 
of  a  point  moving  from  the  effect  of  an  impulsion  only,  then,  as 
there  is  here  no  acceleration,  the  equations  of  the  motion  are 

d'x        d'y        d-z 
dt^       'HF       'dt^        ' 

multiplying  each  by  dt  and  integrating,  we  have,  for  the  velocities 
in  the  directions  of  the  axes,  the  expressions 


dx         dy     .    dz 


dt         dt         dt 
and  therefore  (equation  4,  p.  145,)  the  velocity  along  the  path  is 
n2 


150  ELEMENTS  OF  DYNAMICS. 

ds 

v=x/\a'*+b'+c'^'\  which  is  constant ;  that  is,  — =C  .-.  s=C/  +  C,, 

at 

Cj  being  the  space  already  passed  over  when  /"  commences. 

From  equations  (1)  we  can  prove  that  the  path  of  the  body  must 
necessarily  be  a  straight  line  ;  for,  multiplying  each  by  dt  and  inte- 
grating, they  give  x^=(tt-\-u',  y=bt-\-b',  z=ct-\-c',  by  means  of 
which  eliminating  /,  and  there  results  the  following  general  rela- 
tions among  the  co-ordinates,  viz. 

a       a'c — ac'         b     ,  b'  c  —  be' 
c  c         ^     c  c 

As  a  second  application,  let  it  be  required  to  determine  the  motion 

of  a  projectile  in  space.     Here  the  equations  of  the  motion  are 

d'' X         d'^y 

-J— =0,  —i~=  —  gi  therefore,  multiplying  by  dt  and  integrating, 

we  have  for  the  components  of  the  velocity,  — =C,  -^=C'  —  gt, 

multiplying  by  dt  and  integrating  again,  we  have, 

x=Ct,  y=C't  —  kgt^;  or,  putting  C'=aC, 
y=ax — ^gt",  as  before  found. 


CHAPTER  II. 

ON  THE  MOTION  OF  A  BODY  CONSTRAINED  TO  MOVE  ON  A  Cn'EN  CURVE. 

(125.)  In  this  chapter  we  shall  show  the  application  of  the  fore- 
going general  theory  to  the  circumstances  of  constrained  motion, 
the  moving  body  being  prevented  from  obeying  the  influence  of  the 
applied  forces  through  the  intervention  of  a  rigid  line. 

When  a  material  point  is  thus  compelled  to  move  on  a  curve,  the 
curve  offers  at  every  point  passed  over  a  certain  resistance  to  the 
motion  in  the  direction  of  the  normal,  and  it  is  in  consequence  of 
this  resistance  that  the  wonted  path  of  the  body  is  continually  di- 
verted and  its  motion  confined  to  the  curve.  This  resistance,  there- 
fore, may  be  considered  as  a  normal  force  continually  acting  on  the 
moving  body,  and  which,  combined  with  the  other  forces  on  the 
body,  produces  the  motion  which  actually  has  place.  Omitting  the 
normal  force  and  taking  the  components  X,  Y  of  the  others  on 
which  alone  tlie  velocity  along  the  curve  depends  (p.  149),  we  have, 
by  equation  (2,  p.  146), 

v'=2f{Xdx+Ydy)....(l), 
by  means  of  which  the  velocity  at  any  proposed  point  (x,  y)  of  the 


CONSTRAINED    MOTION.  151 

curve  may  be  found,  X  and  Y  being  functions  of  the  co-ordinates. 
As  shown  at  (p.  146),  the  expression  after  integration  will  take  the 
form  v^  —  '^i^=f{.^i  y) — ^/(«»  ^)^  («>  b)  being  the  point  where  the 
velocity  is  known  to  be  v^. 

As  this  result  is  independent  of  the  normal  pressure,  that  is,  as  it 
remains  the  same  whatever  tliis  pressure  may  be,  we  infer  that  it 
must  remain  the  same  whatever  the  curve  between  (a,  b)  and  {x,  y) 
may  be,  so  that  as  long  as  the  applied  forces  remains  the  same,  and 
the  velocity  of  the  "body  at  the  point  (a,  b)  remains  the  same,  the 
velocity  at  any  other  point  {x,  y)  will  be  the  same  by  whatever  path 
it  arrives  at  it. 

Precisely  the  same  conclusions  would  follow  if  we  had  supposed 
the  motion  to  be  on  a  curve  of  double  curvature  instead  of  on  a 
plane  curve,  as  is  very  obvious  from  the  equation  at  p.  146. 

If  the  body  move  on  the  curve  in  virtue  of  an  original  impulse 
merely,  then,  since  there  are  no  acting  forces,  X=0  and  Y=0,  and 
consequently  v^^v^,  which  shows  that  the  primitive  velocity  will 
be  preserved  and  continued  unchanged  whatever  be  the  curve  along 
which  it  moves. 

Let  us  suppose  the  body  to  move  down  a  curve  in  consequence 
of  the  action  of  gravity,  then  we  have,  by  taking  the  axis  of  X  vertical, 

X=  —  g,  Y=—0.\v^=2fXdx=v^^  —  2gx. 
To  determine  v^  let  h  be  the  height  above  the  origin  from  which 
the  body  begins  to  descend,  that  is,  the  ordinate  of  the  point  at 
which  the  velocity  is  0,  then 

0=v^''~2gh  .'.v^''=2  gh  .'.v^=2  g  {h  —  x)  ....  (2). 

In  the  general  case  of  this  problem  we  have  seen  that  the  velocity 
depends  on  the  co-ordinates  x  and  y  of  the  j)oint  arrived  at,  but  is 
independent  of  the  path  to  it ;  in  this  particular  case  we  see  that 
the  velocity  depends  only  on  the  ordinate  x  of  the  point  arrived  at, 
being  independent  both  of  the  abscissa  of  the  point  and  of  the  path, 
and  thus  any  point  in  a  straight  line  parallel  to  the  horizon  will 
be  arrived  at  ivith  the  same  velocity  if  the  body  descend  from  a 
fixed  point  above  it  along  any  line  or  ciirve  whatever,  the  velocity 
being  that  which  would  be  acquired  by  falling  freely  through  the 
vertical  height. 

It  immediately  follows  from  this,  that  when  the  body  has  arrived 
at  the  lowest  point  of  the  curve,  its  acquired  velocity  will  be  suffi- 
cient to  carry  it  up  the  ascending  branch  (if  the  curve  have  one,)  to 
the  same  height  as  it  descended  from,  whether  the  two  branches  be 
similar  or  not,  although  the  times  of  descent  and  ascent  will  be  dif- 
ferent if  the  branches  be  different  in  form. 

To  determine  the  time  requires  that  we  know  the  curve  of  descent, 
in  which  case  we  have,  by  the  general  expression  (D),  art.  (106),  v=- 


152  ELEMENTS  OF  DYNAMICS. 

(Is  J,       (h      ...       ,  . 

—  .'.  (It  =—,  that  IS,  m  the  case  ol  gravity. 

If  ^'*  p  ds  ,  . 

'  =^\2g{h  —  x)\'''^=J  ^\2g{h  —  x)\  ••••/  >' 

(126.)  "We  shall  shortly  give  an  example  or  two  of  this  kind  of 
constrained  motion ;  but  we  shall  first  investigate  a  general  expres- 
sion for  the  resistance,  or  normal  force,  at  any  point  of  the  con- 
straining curve,  when  this  curve  is  given. 

Let  APM  (fig.  99,)  be  any  given  curve  on  which  a  material  point, 
P  is  compelled  to  move  when  acted  upon  by  forces  whose  compo- 
nents are  X  and  Y.  Let  PN  represent  the  normal  force  or  the  re- 
sistance which  the  body  receives  when  at  P  and  call  it  R ;  the  com- 
ponents of  this  force  are  R  cos.  NPC  and  —  R  cos.  NPC  ;  conse- 
quently, taking  into  account  all  the  forces  which  act  upon  the  body, 
the  equations  of  its  motion  are 

^=Y-R-]    ""^^' 
dt''  da  J 

Multiplying  the  first  of  these  by-p,  and,  the  second  by -p,  and  sub- 
tracting the  second  from  the  first,  we  have 

dyd^x  —  dxd'^y  ^^dy      ^dx 

ds  dt^  ds  ds        ^      ' 

consequently,  since  (DiJ^.  Calc.  p.  136,)  the  general  expression  for 
the  radius  of  curvature  y  at  any  point  (x,  y)  is 


r=-r^Ji j—JT-^  i*  follows  that 

dyd^x — dxd^y 

dx  dy      1   ds^  dx  dy     v^ 

^-^  'ds~^  is'^'^  w-^  rf^    ^  rfi+7  •  •  •  •  ^^^- 

Now  the  expression  X  -^  —  Y  -r-  is  the  result  obtained  by  resolving 

CIS  CIS 

the  forces  X  and  Y  in  the  direction  of  the  normal,  and  which,  there- 
fore, if  the  body  were  at  rest,  would,  when  taken  negatively,  denote 
the  resistance  of  the  curve  ;  but  being  in  motion,  the  curve  suffers 

zn  additional  resistance  expressed  by  — .     We  must  here  remark, 

7 
however,  that  we  have  considered,  in  the  above  reasoning,  the  re- 
sistance R  to  be  offered  by  the  concave  side  of  the  curve  ;  but  if, 
on  the  contrary,  the  body  press  against  the  convex  side,  then  R  will 
not  be  the  sura  but  the  difference  of  the  resistances,  that  is,  the  re- 


CONSTRAINED   MOTION.  153 

sistance  expressed  by  —  must  be  subtracted  from  the  resistance 

'^ 
which  the  curve  would  oppose  if  the  body  were  at  rest.* 

When  the  body  is  retained  on  the  curve  by  the  force  of  gravity 

only,  then  Y=0  and  X= — g\  therefore,  in  this  case, 

^2 

As  the  pressure  =p — ,  at  any  proposed  point,  depends  solely  upon 

the  velocity  at  that  point,  it  would  remain  the  same  if  this  velocity 
were  produced  by  a  primitive  impulse  only,  and  the  motion  to  be 
uninfluenced  by  any  acting  forces,  as  indeed  is  plain  from  the  ex- 

pressionR=  ± — ,  for  the  resistance,  when  X  =0  and  Y=0.     It 

is  readily  seen,  therefore,  that  the  pressure  of  which  we  speak 
arises  entirely  from  the  inertia  of  the  moving  body,  or  its  tendency 
to  move  when  at  any  point  of  the  curve  in  the  direction  of  a  tan- 
gent and  with  its  acquired  velocity ;  this  tendency  necessarily 
causes  it  to  exert  a  pressure  against  the  deflecting  curve,  and  which, 
as  we  have  just  seen,  requires  the  curve  to  oppose  the  resistance 

±  —  m  addition  to  the  resistance  necessary  to  oppose  the  normal 

r 
effect  of  tne  actmg  forces. 

^2* 

A  distinct  name  is  given  to  the  normal  force  or  pressure  =p , 

whose  action  on  the  body  thus  tends  to  repel  it  from  the  centre  of 
curvature  at  that  point  of  its  path  where  its  velocity  is  i; ;  it  is  called 
the  centrifugal  force. 

If  the  curve  on  which  the  body  moves  is  a  circle,  and  if  we  con- 
ceive that  at  the  instant  the  velocity  is  v  an  attractive  force  expressed 

1^2 

by  —  be  placed  at  the  centre,  then,  if  at  the  same   instant  all  the 

other  forces  were  destroyed,  the  body  would  continue  to  move  in 
the  same  circle  and  with  the  same  velocity  v  ;  for  the  repelling  force 
tending  to  increase  the  distance  of  the  body  from  the  centre,  is 
just  balanced  by  the  attractive  force  tending  to  confine  the  body  to 
the  curve,  and  moreover  as  i;,  if  the  body  were  left  to  itself,  would 
continue  unchanged  throughout  the  curve  (p.  151),  the  force  which 

*  The  student  will  not  fail  to  remark  that  the  upper  sign  applies  when  w« 
consider  the  body  to  be  moving  on  the  convexity  of  the  curve,  in  which  case  the 
pressure  due  to  the  acting  forces  is  obviously  diminished  by  this  ;  and  the  lower 
sign  applies  when  the  body  moves  on  the  concavity,  in  which  the  pressure  is 
necessarily  increased  by  the  same  quantity. 

20 


154  ELEMENTS  OF  DYNAMICS. 

counteracts  the  pressure  arising  from  v  at  one  point  of  the  circular 
path  will  be  competent  to  do  so  at  every  point ;  as,  therefore,  all 
pressure  on  the  curve  is  destroyed,  the  motion  of  the  body  cannot 
be  affected  if  the  rigid  curve  were  removed  and  the  body  left  un- 
constrained. 

This  fact  is  at  once  deducible  from  the  expression  (2)  for  R,  and 
indeed  in  a  more  general  form ;  for  in  order  that  R  may  be  0,  wliich 
is  the  same  as  saying  in  order  that  the  motion  may  continue  un- 
changed though  the  resisting  curve  be  removed,  we  see  that  the 
normal  effect  of  the  applied  forces  must  be  equal  and  opposite  to 

the  force  — :  so  that  if  a  force  always  expressed  by  this  were  al- 

y 
ways  to  act  at  the  centre  of  curvature,  corresponding  to  the  radius 
y,  the  rigid  curve,  be  it  what  it  may,  might  be  removed  without 
changing  the  trajectory.  By  this  arrangement  it  is  plain  that  the  va- 

riable  force  —  must  itself  move  so  as  to  describe  the  evolute  of  the 

curve  described  by  the  body ;  the  evolute  of  a  circle  is  a  point,  viz. 
the  centre. 

As  the  central  force,  or  as  it  is  usually  called  the  centripetal 

force,  necessary  to  retain  a  body  in  a  circle  is  F  =     ,  and  since, 

7 
if  /"  be  the  time  of  one  revolution,  we  must  have,  in  consequence 

2  rt  Y  4  rt^Y 

of  the  uniform  velocity,  w  = .•.F  =  — ^,  which  expresses 

alike  the  intensity  of  either  the  centripetal  or  the  centrifugal 
force. 

In  like  manner,  for  any  other  circle  of  radius  y^,  and  time  <,", 
the  centripetal  or  centrifugal  force  is 

F  -ll'll  .  FF  ■■^■^^^ 
r,_    ^^^     ..r  .r,..  ^,  .^^,  , 

hence,  1st.  When  the  circles  are  equal,  the  centripetal  or  centri- 
fugal forces  are  inversely  as  the  squares  of  the  times  ;  and  2d.  When 
the  times  of  revolution  are  equal,  the  forces  are  as  the  radii  of  the 
respective  circles. 

Let  us  apply  these  results  to  an  interesting  problem,  viz.  to 
the  determination  of  the  centrifugal  force  at  different  places  on  the 
earth's  surface,  from  knowing  the  time  of  one  rotation  on  its 
axis. 

The  earth,  by  means  of  its  diurnal  motion,  carries  round  with  it, 
with  a  uniform  velocity,  every  point  on  its  surface  in  86164  se- 
conds. At  the  equator  the  radius  y  is  20921185  feet,  therefore  the 
centrifugal  force  at  the  equator  is 


CONSTRAINED  MOTION.  155 

„      4rtV      4rt=20921185        ,,,„,,     ^ 

As  this  force  opposes  the  force  of  gi'avity,  it  follows,  that  if  it  did 
not  exist,  that  is,  if  the  earth  did  not  revolve  on  its  axis,  the  force 
of  gravity  instead  of  being  what  it  really  is,  viz.  g':=32-08818  at 
the  equator,  would  be  G  =  g"  +  •1112447,  and  thus  the  Aveight  of 

•1112447  ,       . 

any  body  would  be  a       noof^"  P^""^  more  than  it  really  is. 

32^08818 

The  ratio  of  G  to  F  being 

32  •  1994247  :  •I  112447  or  289  :  1  nearly, 

.■.F^^....m. 

Now  every  parallel  to  the  equator  being  carried  round  in  the  same 
time  t"  as  the  equator,  we  have,  by  representing  the  centrifugal 
force  in  the  parallel  whose  latitude  is  I  and  radius  y,  by  F^, 
r:T.  ::F:F, 

because  it  is  evident  that  yj=y  cos.  /. 

The  force  G  of  gravity  is  not  diminished  by  the  whole  of  the 
centrifugal  force  F^,  except  at  the  equator,  because  this  force  acts  in 
any  parallel  PAP'  (fig.  100,)  not  in  the  direction  Fp  opposite  to 
gravity,  but  in  the  direction  Pr,  if,  therefore,  we  decompose  the 
force  Pr  in  the  perpendicular  directions  Pjo,  Fq,  both,  as  well  as 
Pr,  being  in  the  plane  of  P's  meridian,  we  shall  have,  for  the  force 
Pp  opposing  gravity, 

P;?=Pr  cos.p'Pr=F^  cos.  POQ=F,  cos.  /; 
hence  the  expression  for  the  diminution  of  gi'avity  is  (equa.  2,) 

-^cos.^/; 

which  therefore  varies  as  the  square  of  the  cosine  of  the  latitude. 

The  other  force  Fq,  being  tangential,  tends  to  draw  the  particles 

of  the  revolving  body  from  the  poles  towards  the  equator,  and  to 

cause  it  to  assume  the  figure  of  an  oblate  spheroid ;  the  expression 

for  this  force  is 

G  C 

Fq=F^  sin.  ^=-^g^sin.  /  cos.  Z=— — sin.  2  I, 

which  therefore  varies  as  the  sine  of  twice  the  latitude. 

From  the  foregoing  principles,  let  it  now  be  required  to  deter- 
mine the  time  in  which  the  earth  must  perform  its  diurnal  revo- 
lution in  order  that  the  centrifugal  force  at  the  equator  may  be  ex- 
actly equal  to  the  force  of  gravity,  or  in  order  that  a  body  may  have 
no  weight  there. 


156  ELEMENTS  OF  DYNAMICS. 

Let  F,  represent  the  time  of  rotation,  corresponding  to  which  the 
centrifugal  force  is  G,  then,  as  tlie  centrifugal  forces  in  the  same 
circle  are  inversely  as  the  squares  of  the  times,  we  have,  (equa.  1,) 

Hence  if  the  diurnal  rotation  of  the  earth  were  performed  in  a  17th 
part  of  the  time  really  occupied,  or  if  it  were  to  turn  round  17  times 
as  rapidly,  bodies  at  the  equator  would  lose  all  their  w^eight,  and 
would,  therefore,  if  placed  at  a  small  distance  above  the  surface, 
remain  suspended  without  any  visible  support ;  if  the  earth  were  to 
revolve  still  more  rapidly  than  this,  no  body  could  remain  on  its 
surface :  every  thing  would  be  repelled  from  it  by  the  centrifugal 
force.  This  inference  it  must  be  remembered  is,  as  well  as  that 
implied  in  equation  (2),  on  the  supposition  that  the  earth  retains  its 
spherical  figure  during  its  rotation,  which  however  is  not  strictly 
correct  on  account  of  the  oblique  influence  of  the  centrifugal  force 
tending  to  elevate  the  equator  and  to  depress  the  poles,  thus  giving 
to  the  earth  a  spheroidal  form.  It  is  demonstrated  that  a  fluid 
spheroid  of  the  same  density  as  that  of  the  earth,  cannot  remain 
in  equilibrium  if  it  revolve  in  a  shorter  time  than  ^'  25'  26".* 

(126.)  The  general  equations  (2)  and  (3),  at  page  151,  contain 
all  that  is  necessary  for  the  determination  of  the  motion  of  a  body 
down  any  given  curve  by  the  action  of  gravity;  or  indeed  of  any 
force  g  acting  in  parallel  lines.  The  theory  of  the  pendulum,  a 
highly  important  subject,  is  established  by  their  aid,  and  to  this 
theory  w^e  shall  apply  them  in  the  following  chapter.  We  may  re- 
mark, however,  before  entering  upon  this,  that  the  expression  (1) 
for  the  velocity  at  page  150,  is  general  for  all  hypothesis  of  the 
acting  forces,  and  when  this  velocity  is  determined  and  the  curve 

ds 
given,  the  time  will  be  given  by  the  equation  t  =f — ,  in  the  par- 
ticular case,  however,  where  the  body  is  acted  upon  by  a  single 
centre  of  force,  varying  according  to  some  function  of  its  distance, 
then,  for  the  determination  of  the  velocity,  it  will  be  proper  to  use 
the  expression  (1)  at  page  147,  taking  the  integral  with  a  negative 
sign  if  the  force  be  attractive,  tending  to  diminish  the  co-ordinates, 
and  taking  it  with  a  positive  sign  if  repulsive ;  the  axes  of  reference 
too  must  here  originate  at  the  centre  of  force. 

*  See  Professor  Airy''s  Tracts,  iiage  150,  second  edition;  or  the  Thiorie 
Analytique  du  Systeme  du  JMonde,  of  Poiit^coxilant ;  torn,  ii.,  page  400. 


ON  THE  SIMPLE  PENDULUM.  157 


CHAPTER  III. 


ON  THE  SIMPLE  PENDULUM. 


(127.)  A  SIMPLE  pendulum  is  considered  to  be  a  material  point, 
attached  to  a  thread  or  rod  without  weight,  and  oscillating  about  a 
fixed  axis  connected  with  the  other  extremity  of  the  rod.  Such  a  pen- 
dulum, it  is  evident,  can  have  no  physical  existence,  yet  it  is  con- 
venient to  discuss  the  theory  of  such  an  imaginary  pendulum,  be- 
cause, as  will  be  shown  in  a  subsequent  chapter,  whatever  be  the 
oscillating  body,  there  may  always  be  found  a  point  at  which,  if  a 
single  particle  were  placed  and  connected  by  a  rod,  without  weight, 
to  the  point  of  suspension,  the  oscillations  of  this  simple  pendulum 
would  be  performed  in  the  same  time  as  those  of  the  compound 
body ;  and  all  the  circumstances  of  its  angular  motion  would  be  the 
same,  and  thus  any  pendulum  may  be  reduced  to  an  equivalent 
simple  pendulum. 

The  moving  point  which  we  here  consider,  is  confined  to  the 
curve  in  which  it  moves  by  the  thread,  the  accelerating  force  being 
gravity ;  hence,  the  tension  suffered  by  the  string  at  any  point  of 
the  path,  must  be  equivalent  to  the  pressure  which  would  be  sus- 
tained by  the  curve  at  that  point  if  it  were  rigid,  and  the  moving 
point  were  unconnected  with  the  thread.  The  constraining  forces 
being  equivalent,  the  theory  developed  in  the  preceding  chapter  be- 
comes immediately  applicable  to  the  motions  of  simple  pendulums ; 
these  motions,  although  usually  in  circular  arcs,  may  nevertheless 
oe  in  any  curves  whatever,  for  the  thread  as  it  oscillates  to  and  fro 
may  be  forced  to  wrap  itself  about  curves  springing  from  the  point 
of  suspension,  and  thus  the  material  point  will  be  forced  to  describe 
curves  which  are  the  involutes  of  these ;  and  in  this  way  we  may 
have  circular  pendulums,  cycloidal  pendulums,  &LC.  Of  these,  how- 
ever, the  circular  pendulum  is  the  most  simple  and  important. 

(128.)  It  wiU  contribute  to  the  convenience  of  the  student  to 
bring  together  in  this  place  those  formulas  distributed  in  the  pre- 
ceding chapter,  which  are  required  in  the  problems  we  are  about 
to  give ;  these  formulas  are  as  follow : 

velocity  =  v  =  \/  ^g(h — x)'  ♦  •  •  (A), 

/»       ds*  ,^^ 

time  =  ;  ==/ —  (B), 

J  s/%g{h—x) 

*  If  we  consider  the  arc  s  in  this  expression  to  be  the  arc  of  descent,  this, 
being  measured  from  the  origin  or  lowest  point,  must  diminish  as  the  time  in- 

o 


158  ELEMENTS  OF  DYNAMICS. 

centrifugal  force  =/= — =— ^^ .  .  .  .(C), 

y  y 

_        dy     2sr(h  —  x)     ,^,  du  . 

tension  =  T  =  ^-^H — ^— ^ •  (D),  or,  since  -p  expresses  the 

cosine  of  the  angle  which  the  normal  makes  with  the  axis  of  x,  if 
we  call  this  angle  a  we  may  write  the  last  expression  thus  : 

2  o-  (// — x) 
tension  =  g-cos.  a  -\ — ^— ^^ (E), 

y 
which  form  will  be  sometimes  most  convenient  to  use,  (see  prob.  IV. 
following.) 

Problem  I,  (129.)  To  determine  the  time  of  oscillation  in  a  cir- 
cular pendulum  (fig.  101). 

Taking  the  origin  of  the   vertical    and    horizontal    axes  at  the 
lowest  point  of  the  curve,  and  calling  the  radius  of  the  circle  or  the 
length  of  the  rod  r,  we  have,  for  the  equation  of  the  path, 
-     «  .        dy""       (r  —  x)' 


.*.   dS: 


dx^      2rx — x^ 
dy^      ,  rdx 


rfx2*  ^\  2rx—x^\ 

consequently,  by  equation  (B)  above,  the  expression  for  the  time  is 

/ds  _      **         /*  d""^ 

VVig(ih—^)'~V\2f\  J  :yj{h:^{2rx—x^)l' 
The  differential  expression  under  the   integral  sign   may  be   put 

under  the  more  convenient  form  — ; — ^ -; r^(l — :r-)~^  5 

^\2rl^  \nx — x^p        2r 

which  shows,  that  if  the  second  factor  be  developed  by  the  binomial 

theorem,  the  difTerential  in  question  will  be  reduced  to  a  series  of 

3?"  dx 

others  all  of  the  form  — rr r.  which  we  know  to  be  an  in- 

^\hx — x^\ 

tegrable  form. 

These  details  we  have  entered  into  at  length  in  the  Integral  Cal- 
culus, page  93,  and  the  result  is,  that  the  proposed  integral,  taken 
between  the  necessary  limits,  that  is  from  x  =0,  the  lowest  point 
of  the  curve,  to  x=h,  the  point  of  departure,  is 

creases,  and  therefore  ds  in  this  formula  is  negative ;  but  if  we  refer  to  the  arc 
of  ascent,  the  time  and  arc  increase  together,  and  ds  is  positive,  we  shall  there- 
fore always  consider  the  formula  ais  applied  to  the  ascending  arc,  since  the  time 
will  be  the  same  whether  the  body  descend  from  the  height  x  to  the  lowest  point, 
or  ascend  from  the  lowest  point  to  the  height  .r  ;  or  whether  it  descend  from  h  to 
X,  or  ascend  from  x  to  h.  On  this  hypothesis,  therefore,  the  integral  (B)  com- 
mences at  x^=x  and  ends  at  x=h ;  for  the  descending  arc,  on  the  contrary,  the 
integral  commences  at  x=A  and  ends  at  h=x. 


ON    THE    SIMPLE    PENDULUM.  159 

dx 


f 


y/\(Ji  —  x)  {2rx  —  x^)\ 


consequently  for  2  T',  or  the  time  of  a  complete  oscillation,  we  have 

^  ^*     ^^2^  2r^^2.4^  hr^    ^    "^■<^:!, 

by  means  of  which  the  time  may  be  approximated  to,  to  any  degree  of 
accuracy.     The  expression  h  is  the  versed  sine  of  the  arc  of  descent, 

or  of  half  the  whole  path,  and—  is  the  versed  sine  of  a  similar  arc 

r 

to  radius  1,  and  therefore  of  the  inclination  of  the  rod  to  the  verti- 
cal in  its  initial  position ;  the  smaller  this  inclination  is  the  more 
convergent  will  the  foregoing  series  be.  Suppose,  for  instance,  the 
initial  inclination  were  5°,  then  the  versed  sine  of  this  being  "0038053, 
the  second  term  of  the  series  would  be  only 

,  1  ,  -0038053       ^^„,    , 
(-)« 2 =-0004757, 

and  if  the  pendulum  vibrated  but  one  degree  on  each  side  of  the 

vertical,  then,  since  the  versed  sine  of  1°  is  '0001523,  the  second 

,,  ,     ,      ,  1    ,     -0001523 
term  of  the  series  would  bebut(— -  y =  -000019,  and, 

Ai  Ai 

by  supposing  the  arc  of  vibration  less  and  less,  the  expression  for 
the  time  of  an  oscillation  would  continually  approximate  to 

2f==rtv/—  ••••(2), 
E 
which  will  differ  insensibly  from  the  true  time  when  the  arcs  of  vi- 
bration are  very  small,  that  is,  not  exceeding  about  4°  ;  and  there- 
fore, for  all  arcs  between  this  and  0,  the  times  of  vibration  of  the 
same  pendulum  will  not  perceptibly  differ,  that  is,  in  very  small 
arcs  the  oscillations  may  be  regarded  as  isochronal,  or  as  all  per- 
formed in  the  same  time. 

Since  the  time  occupied  by  a  body  falling  freely  through  the  height 

^  r  is  expressed  by^— ,  it  follows,  from  the  foregoing  expression, 

that  in  pendulums  of  such  limited  ranges  or  amplitudes  as  we  have 
supposed,  the  time  of  vibration  is  to  the  time  of  falling  freely 
through  half  the  length  of  the  rod  a*  3-14159  fo  I. 


160  ELEMENTS  OF  DYNAMICS. 

It  is  an  important  matter  to  know  exactly  the  length  of  a  pendu- 
lum wliich  will  vibrate  seconds,  and  combined  with  experiment  the 
the  f^oneral  expression  (1)  will  cnal)le  us  to  determine  this  length 
with  perfect  accuracy  for  any  given  arc  of  vibration. 

Thus,  let  r,  r'  be  the  lengths  of  two  pendulums  vibrating  in  arcs 

h     h' 
of  the  same  number  of  degrees,  then,  since  —  =  — the  series  within 

r       r 

the  brackets  will  be  the  same  for  each  pendulum  ;  hence,  the  times 

of  oscillation  of  these  pendulums  will  be  to  each  other  as 

r  r' 

ftx/—  to  }t^ —  or  as  ^r  to  >/r': 
g  g 

the  times  of  oscillation  are  therefore  as  the  square  roots  of  the 
lengths.  Let  the  pendulum  r  make  n  oscillations  in  the  same  time 
t"  that  the  pendulum  r'  performs  n'  oscillations,  then  the  respective 

t"        t" 
times  of  a  single  oscillation  will  be  —  and—,  which  are   to  each 
®  n         n 

other  as  —  to  —  ;  hence,  by  the  proportion  just  deduced, 

r  :  r   ::  — -  :  — —  : :  n "  :  7i', 

that  is,  the  lengths  of  pendulums  vibrating  in  similar  arcs  are  to 
each  other  inversely  as  the  squares  of  the  num,ber  of  oscillations 
made  by  them  in  the  same  time. 

Now  a  seconds'  pendulum  must  vibrate  t  times  in  t",  if,  therefore, 
we  take  a  pendulum  of  any  length  r,  and  count  the  number  n  of  vibra- 
tions it  makes  in  any  time  t",  we  shall  find  the  exact  length  r'  of  the 
seconds'  pendulum  vibrating  in  a  similar  arc  by  this  proportion,  viz. 

n^  r 
t':n'::r:r'  =—  ....  (3). 

If  the  length  of  the  seconds'  pendulum  be  thus  determined  for  very 
small  arcs,  we  may  thence,  by  help  of  the  expression  (2),  determine 
the  force  of  gravity  at  the  place  where  the  experiment  is  made  for,  as 

l=Hy-.:g=^^r'-i4). 

Now  in  the  latitude  of  London  r'  has  been  found  to  be  =:39  .  14 
inches,  consequently  ^=7t^x39  .  14  in. =32*19  feet,  the  force  of 
gravity  in  the  latitude  of  London. 

Knowing  the  length  of  the  seconds'  pendulum,  it  will  be  an  easy 
matter,  from  the  foregoing  theorems,  to  find  the  time  of  vibration  of  a 
pendulum  of  any  other  length,  or  the  lengUi  of  a  pendulum  vibrating 
in  any  other  time.  Thus,  the  length  of  the  seconds'  pendulum  being 
r',  and  that  of  any  other  r,  we  have  by  the  first  of  those  theorems, 
this  expression  for  the  number  t  of  seconds,  this  last  will  vibrate  in, 


ON    THE    SIMPLE    PENDULUM.  161 

viz.   t  =      f— ,  •••  r  =  r'  t^.      Suppose,  for  example,  we  wanted 
to  know  the  time  of  an  oscillation  of  a  pendulum  20  feet  long, 


we  should  then  have  t  =     f— - — — -  =  2.5  nearly,  so  that  the  time 
^yoU  .  14 

would  be  about  2  seconds  and  a  half. 

Again,  if  we  wanted  to  know  the  length  of  a  pendulum  that 
should  oscillate  once  in  ten  seconds,  then  we  have 
r=39  .  14x10^=3914  .  inches. 
We  may  also  readily  determine  the  number  of  seconds  lost  or  gain- 
ed in  a  day  by  lengthening  or  sliortening  a  seconds'  pendulum  by 
any  proposed  quantity ;  for,  from  the  equation  (3),  we  have 

_r'  P 

where  r'  is  the  length  of  the  seconds'  pendulum,  and  f =86400,  the 
number  of  seconds  in  24  hours,  n  being  the  number  of  times  the 
altered  pendulum  r  oscillates  in  24  hours.  Suppose  r=r'-{-p,  and 
n=t  —  q,  p  will  then  be  the  error  in  length  of  the  altered  pendu- 
lum, and  q  the  consequent  deficiency  in  the  number  of  vibrations, 
or  the  loss  in  seconds,  and  we  shall  have 

r  +p=  -^ =r  +2r  — +  &c. 

^^       t2—2tq-\-q"-  ^        t 

If  the  loss  amount  to  but  a  few  seconds,  the  powers  of  —  may  ob- 
viously be  neglected  without  sensible  error,  and  we  shall  thus  have 
;j=2r^  -^  and  0=  -r — 

the  first  equation  showing  the  increase  of  length  corresponding  to 
a  given  loss,  and  the  second  showing  the  loss  consequent  upon  a 
given  increase  of  length ;  and  the  expressions  hold  when  the  pen- 
dulum is  diminished  hyp;  q  then  expressing  the  gain. 

Hitherto  we  have  considered  the  pendulums  compared  to  oscil- 
late at  the  same  place  ;  but  it  is  a  very  important  inquiry  to  deter- 
mine the  lengths  of  pendulums  oscillating  seconds  at  different  places 
on  the  earth's  surface,  as  such  a  determination  readily  leads  to  the 
discovery  of  the  true  figure  of  the  earth.  If  we  represent  by  G  and 
g  the  intensities  of  gravity  at  any  two  places,  and  by  r  and  r'  the 
lengths  of  the  corresponding  seconds'  pendulums,  then,  by  equation 

G  £• 

(4),  we  shall  have  >'=— ,  »''=-^»  so  that  the  intensity  of  gravity 

at  any  places  varies  as  the  length  of  the  seconds''  pendulum  at 
those  places.     But  it  is  manifest  that  the  intensity  of  gravity  must 
0  2  21 


162  ELEMENTS    OF    DYNAMICS. 

depend  upon  the  figure  and  constitution  of  the  earth,  and  accord- 
ingly it  is  proved,  by  the  writers  on  Physical  Astronomy,  that, 
considering  the  earth  to  be  a  homogeneous  spheroid  of  equilibrium, 
the  intensity  of  gravity  must  vary  as  the  normal,  so  that,  from  the 
foregoing  equations,  the  normal  to  the  earth's  surface  at  any  place 
varies  as  the  length  of  the  seconds'  pendulum  at  that  place  ;  aud 
thus  when  the  lengths  of  the  pendulum  for  any  two  known  latitudes 
are  accurately  ascertained  by  experiment,  sufficient  data  will  be 
furnished  for  determining  the  ratio  of  the  earth's  polar  and  equato- 
rial diameters,  or  for  finding  the  ellipticity  or  spherical  compression, 
as  it  is  called,  and  by  which  is  meant  the  ratio  of  the  difference  of 
these  two  diameters  to  the  greater.  But  we  shall  make  the  deter- 
mination of  this  ratio  from  the  proposed  data  a  distinct  problem. 

Problem  II. — (130.)  To  determine  the  compression  or  ellipti- 
city of  the  earth  by  means  of  seconds'  pendulums. 

Let  a,  b  represent  the  equatorial  and  polar  semi-diameters,  e  the 

eccentricity,  and  c= ,  the  compression;  then 

a^  —  h^     a  —  b     a-\-b 

e»= — -  = X— !— =c    2  — c  . 

cr  a  a 

Now  c  is  itself  but  a  small  fraction,  and  the  square  of  it  is  too  small 
to  be  worth  regarding  in  this  inquiry,  so  that  we  may  consider  the 
compression  to  be  expressed  by  c=k  t^. 

Let  X,  /t'  be  the  latitudes  at  which  the  lengths  of  the  seconds' 
pendulums  are  /,  /',  then,  since  (Diff.  Calc.  p.  134,)  the  expres- 
sions for  the  normals  at  these  latitudes  are 

6''  1  ,      ^^  i 

^=  T  '  (1  — c'^sin.U)!'  ^  ""  "^  ■  (1— e'^sin.U')!  ' 
we  have  /:/'::(l  —  e*  sin.  ^  A,)~l :  (1  —  e'^sin.  ^  A-')~i- ,   that  is, 
by  expanding  the   two  last  terms  by  the  binomial  theorem,   and 
omitting  the  square  and  higher  powers  of  e^  sin. "  A,  on  account  of 
their  excessive  sraallness, 

/:/'::  1  +  |  e*sin.  =A:  \-\-h  e^sm.^X' 
::l+csin.  =»A      :l  +  csin.  »A' 


/sin. ''A'  —  /'sin. -A     '    ■     ,  •> ,        •     „, 
— sm.  ^  A — sm.°A 

which  expression  would  be  the  value  of  the  compression,  if,  as  we 
have  supposed,  the  earth  were  of  uniform  density.  Such,  how- 
ever, is  not  the  case,  yet  the  conclusion  just  obtained  will  enable 
us  to  deduce  the  true  compression,  whatever  be  the  law  of  the 


ON    THE    SIMPLE    PENDULUM.  163 

earth's  density,  by  the  aid  of  the  following  very  remarkable  propo- 
sition discovered  by  Clairaiit,  viz.  "  Whatever  be  the  law  of  the 
earth's  density,  if  the  elliptieity  of  the  surface  be  added  to  the  ratio 
which  the  excess  of  the  polar  above   the  equatorial  gravity  bears 

to  the  equatorial  gravity,  their  sum  will  be—-,  m  being  the  ratio 

of  the  centrifugal  force  at  the  equator  to  the  equatorial  gravity."* 
Now  the  ratio  which  the  excess  of  the  polar  above  the  equatorial 
gravity  bears  to  the  equatorial  gravity,  is  no  other  than  the  ellipti- 
eity c,  as  determined  upon  the  hypothesis  of  homogeneity  ;  for, 
calling  the  polar  gravity  G  and  the  equatorial  gravity  g,  and  re- 
collecting that  these  are  as  the  normals,  we  have 

G  :  s^::b:~.-.  ^— =c; 

a  g  a 

consequently,  whatever  be  the  law  of  the  earth's  density  if  we 
call  the  elliptieity  or  compression  C,  we  have 

C  +  c=— -=—  •  — —  (page  109,)  and  therefore 


I 

c= 


1-F 


578        i     .      ..,         .     ,^ 
—  sm.  "/,  —  sm.  ^A 


From  experiments  at  Madras  A=13°  .  4'  .  9"  ,  1=39  •  0234 
From  experiments  at  Melville  Island 

A'=74°  .  47'  .  12"  ,  l'=39  •  2070  , 
5 
from  which  c=-0053478  and  as =-0086505,  therefore 

C  =  -0033027=— . 
302 

From  this  value  it  appears  that  the  equatorial  diameter  of  the  earth 

exceeds  the  polar  by  about  the  300th  part  of  its  whole  length ;  that 

is,  these  diameters  are  to  each  other  as  300  to  299,  and  this  ratio 

agrees  almost  exactly  with  the  ratio  as  determined  by  means  of  the 

actual  measurement  of  degrees.  See  Dr.  Gregory'' s  Trigonometry, 

page  231 ;  and  Mry's  Tracts,  page  186. 

Problem  III. — (131.)  To  determine  the  time  of  oscillation  of  a 
cycloidal  pendulum  (fig.  102). 

When  the   axes  originate  at  the  extremity  of  the  base  ot  the 

*  See  Professor  Airy's  Tracts,  p.  174. 


164  ELEMENTS    Ot    DYNAMICS. 

cycloid  we  have  found  for  the  length  of  any  arc  s  (Int.  Calc.  p. 
115,)» 

s=4;-  — 2^j2r(2r  — a-); (1); 

but  if  we  measure  s  from  the  vertex,  tlion,  since  the  length  of  the 
semirycloid  is  4  r,  we  sliall  have,  by  subtracting  the  foregoing  ex- 
pression from  this,  and  then  removing  the  origin  to  the  vertex,  that  is, 
substituting — x-f  2r  for  x,  we  have,  in  the  inverted  cycloid  (fig.  103,) 

s=2y/^^.:  ds=  J^dx  .'.  t=r — 

\x  J  ^2g  (h—x) 

r  n        (Ix  r  .    _  2      ,  ^ 

=  v/ — / =  a/ — versm.    ^-pX+C; 

gJ    ^  hx—x''         S  " 

I  r  2  . 

hence,   from  x=x   to  x=h,   /=     — In — versin.~*^-x I ,  which 

\^  A 

expresses  the  number  of  seconds  in  descending  from  the  altitude  h 
to  the  altitude  x ;   therefore  the  time  of  descent  to  the  lowest  point 

r 
A  at  which  a:=0  is  t^n^/—  ....  (2). 
g  ' 

This  expression  being  independent  of  h,  is  very  remarkable,  in- 
asmucli  as  it  proves  that  the  time  of  descent  to  the  lowest  point  is 
always  the  same  from  whatever  point  in  the  curve  the  body  be- 
gins to  descend.  The  oscillations  in  a  cycloid  are,  therefore,  al- 
ways isochronal. 

The  cycloidal  pendulum  must  oscillate  between  the  two  equal  cy- 

cloidal  cheeks  SB,  SO,  about  which  the  thread  SP  wraps  itself;  the 

length  of  the  pendulum  being  equal  to  that  of  the  curve  SB,  or  which 

is  the  same,  of  the  semi-cycloid  BA,  and  this  from  equation  (1),  is 

by  putting  y=2r,  s  =  4r,  so  that  calling  the  length  of  the  pendulum 

/  the  expression  for  a  complete  vibration  is,  from  equation  (2), 

/ 
2t=7t^ ; 

or 

O 

which  is  the  same  as  the  expression  given  in  last  problem  for  the 
time  of  vibration  of  a  pendulum  of  the  same  length,  in  a  very  small 
circular  arc. 

Problem  IV. — (132.)  When  a  body  vibrates  in  a  circular  arc,  to 
determine  the  tension  of  the  string  at  any  point  (fig.  101). 

•  An  obvious  error  has  crept  into  the  formula  here  referred  to ;  instead,  of 

s=^'irf~^w^-\-  c 

it  should  have  been 


v^/; 


=— 2v'2r(2r  — y)4-C, 


^ir—y 
and  C  is  determined  from  the  condition  that  »=0  when  y=0. 


ON    THE  SIMPLE    PENDULUM.  165 

Here  the  cosine  of  the  angle  »,  or  PXA,  which  the  normal  makes 
with  the  axis  of  X,  is  obviously ; — .     Hence,  by  the  formula  for 

the  tension,  we  have 

r  —  X    ,    2^(A  —  x)  r  +  2/t  —  3x 

tension  =  2* ^^-^ '-  =  2r 

^      r  r  *  r 

at  the  lowest  point,  or  where  a;=0  tension  =«• ;   which,  if 

the  body  fall  from  /j=r,  becomes  3g ;  that  is,  the  body,  when  it 

comes  to  A,  is  acted  upon  by  three  times  as  much  force  as  it  would 

be  if  at  rest  there,  and,  therefore,  the  body  stretches  the    string 

with  three  times  its  weight.     The  tension  is,  obviously,  greatest  at 

this  point. 

In  order  to  determine  the  point  at  which  the  tension  is  0,  we 

r-\.2h  —  3x       ,^  r-{-2h      ,      ,     . 

have  g =  0  .•.  x  = — - — ,  the  abscissa  of  the  point 

at  which    the  pendulum,  let  fall  from   the  height  h,  produces  no 
strain  on  the  point  of  suspension. 

To  determine  the  point  at  which  the  strain  is  the  same  as  when 
the  body  hangs  at  rest,  we  must  equate  the  expression  for  the  ten- 
sion with  g,  we  thus  find  x=^h. 

Problem  V. — (133.)  To  determine  centrifugal  force  and  the  ten- 
sion of  the  string  in  the  cycloidal  pendulum. 

We  have  already  seen  (prob.  H.)  that  in  the  cycloid  the  expres- 

„       .  dx  .    dx  X 

sion  tor  sin.  a,  or  -^,  is  -7-  =  »/  -- 

ds        ds  2r 


J 2  r—x 
X 


dy       ,,,         .    „,  hr — X      dy  2  r  — 

.-.  COS.  a  or-#  =  v/U  —  sin. ''I  a  =     .-.  ,-=  .  p 

ds     ^  '  ^         \1     2  ?•  dx     S       2r 

dy 

'  '  dx 
By  means  of  these  expressions  we  may  find  the  radius  of  curvature 
at  any  point  (^x,  y,)  from  the  formula,  Diff.  Calc.  p.  124-5.  It  is 
_        ds^   _  d^y  _       2  r  I  — r 

'^~~~dx^~  d3^~'~^'x'^  ~Xs/\xJi~r^x)\ 
^__  =2^  [2r(2r  —  x); 
hence  the  expression  for  the  centrifugal  force  is 
f=—=        g  {h—x)        . 
•^        r        ^\2r{2r—x)\' 


and,  for  the  tension,  we  have 


\2t—x\_^         g(h--x) 


2r        '^  ^  \2  r  {2  r—x)  i' 


166  ELEMENTS    OF    DTN'AMICS. 

At  the  lowest  point,  or  where  x=0,  both  the  centrifiigal  force 
and  tension  are  obviously  greatest ;  the  expressions  for  them  being, 

•/  -  2r  '         -"  +   2r 

Problem  VI. — (131.)  To  determine  the  time  of  gyration  of  a 
conical  pe.dulum. 

When  tlie  pendulum,  instead  of  vibrating  in  a  vertical  plane,  is 
made  to  pass  over,  or  generate,  a  conical  surface,  as  in  fig.  104,  it 
is  called  a  conical  pendulum. 

The  motion  of  such  a  pendulum  is  due  to  three  forces,  viz.  the 
tension  /,  of  the  string  AC  ;  the  force  of  gravity  g,  in  the  direction 
AD,  and  the  centrifugal  force/,  in  the  direction  AB  ;  and  these  three 
forces  keep  the  body,  A,  at  the  same  constant  distance  r  from  S ; 
hence,  resolving  the  tension  t  in  the  directions  AS,  AE,  we  have 

t  COS.  a=/=— (art.  12G),  t  sin.  a=^  ;  therefore, 

_,^  sin.  a      gr      a       gr        ^      gr" 

puttmg  CS  =  a  ;  =  ^ot~  =  —,.\ tJ«=-5— , 

^        ^  cos.  a      v*       r      V*  a 

but  t"  being  the  whole  time  of  gyration,  we  have 

V'  =  —7, — =  -^  •••  f =2  rtj-  ; 
/^  «  \j  g 

which  expression,  being  independent  of  r,  shows  that  the  periodic 

time  varies  as  the  square  root  of  the  altitude  of  the  conical  surface 

described,  whatever  be  the  length  of  the  pendulum,  or  the  radius  of 

the  base  of  the  cone. 

We  have  seen  (prob.  I.)  that  the  time  in  which  a  pendulum  of 

length  a  vibrates  in  a  small  circular  arc  is  expressed  by  t 

hence  the  time  of  gyration  of  a  conical  pendulum  is  exactly  double 
the  time  in  which  a  simple  pendulum,  whose  length  is  the  height  of 
the  cone,  would  vibrate  in  a  small  circular  arc. 


""^7'' 


CHAPTER  IV. 


ON   CENTRAL  FORCES. 


(135.)  The  motions  of  bodies,  acted  on  by  central  forces,  is  a 
branch  of  the  general  theory,  of  so  much  importance,  in  the  system 
of  the  world,  that  it  will  be  proper  to  give  it  a  distinct  considera- 


ON    CENTRAL    FORCES.  167 

tion,  and  to  present  the  equations  of  motion  with  no  more  generality 
than  may  be  requisite,  in  order  to  comprise  the  theory  of  a  body's 
motion,  when  urged  by  a  single  centre  of  force. 

It  will  be  convenient  here  to  put  aside  the  use  of  rectangular  co- 
ordinates, and  to  employ  polar  co-ordinates  originating  at  the  centre 
of  force,  so  that  P  (fig.  105,)  being  the  place  of  the  body  at  any 
time  t"y  and  S  the  centre  of  force,  the  point  P  will  be  determined 
from  knowing  SP=r,  and  the  angle  PSX=u. 

The  law  of  force,  supposed  to  be  some  function  of  r,  will  be 
known  when  the  general  relation  between  r  and  w,  independently  of 
t",  is  known,  and,  conversely,  this  general  relation,  or  the  equation 
of  the  orbit,  will  be  known,  when  the  law  of  force  is  known ;  this 
we  shall  now  proceed  to  show. 

Let  us  suppose  the  force  R  to  be  attractive,  then  we  know  (121) 

ds  ds^ 

that  the  velocity  -7-,  in  the  orbit,  will  be  -^  =  —  2fUdr  .  .  .(1). 

Now.  whatever  be  the  independent  variable,  it  is  shown  (Int.  Calc, 
p.  118)  that  {dsY=r''  {d^y-\-{drY ;  let  then  t  be  the  independent 
variable,  and  we  have,  from  (1), 

dJ^      dr^  ^  .^  J  .^x 

We  moreover  know  (Int.  Cede.  p.  131)  that  the  polar  expression 
for  the  area  generated  in  t",  viz.  the  area  XSP  is  ifr^du,  and  we 
have  seen  (122)  that  this  area  is  proportional  to  t,  that  is, 

fr^do^=ct.'.r^~=c (3). 

As  «  is  the  angular  space  passed  over,  -7-  expresses  the  angular 

velocity,  and  the  equation  (3)  shows  that  this  angular  velocity  varies 
inversely  as  the  square  of  the  distance  of  the  body  from  the  centre 
of  force.     If  we  substitute  in  (2)  the  value  of 

-,  in  (3),  we  have  —+'-=-2fRdr=:v-  .  (4). 

This  equation,  as  it  contains  only  the  two  variables  r  and  t,  which 

we  see  are  at  once  separable,  will  enable  us,  when  R  is  given,  to  find 

the  general  relation  between  r  and  t,  and  thus  the  distance  of  the 

body  from  the  centre  at  any  given  time. 

Again,  by  the  same  two  equations,  viz.  (2)  and  (3),  we  may  eh- 

df 
minate  r  and  — ,  when  ("Rdr  is  found,  and  we  shall  thus  have  a  dif- 

dt  •' 

ferential  equation,  involving  only  o  and  t,  from  which  the  general 
relation  between  «  and  t  may  be  determined ;  and  thus  the  position 
of  the  body  at  any  time  completely  found.     The  two  general  ex- 


168  ELEMENTS    OF    DYNAMICS. 

pressions  for  /,  thus  obtained,  will,  when  put  equal  to  each  other, 
obviously  represent  the  path  of  ihe  body.  But  this  will  be  better 
done  by  eliminating  at  once  (It*  from  the  equations  (2)  and  (3),  by 
which  we  get  for  tlie  diffeiential  equation  of  the  orbit 

c'      1    rfr* 

—.   l-r. -rrr+M  =  — 2/"R(/r=r2  ....  (5) ;   which  will  become 

somewhat  simplified  by  putting  u  for  — ,  as  it  then  takes  the  form 

or  which  is  the  same  thing,  v^-=c^\-^---\-u'^  \  .  .  .  .  (7.) 

—  f/M  has  been  per- 
formed, is  the  difTerential  equation  of  the  orbit,  which,  by  integra- 
tion, will  become  the  algebraical  equation  of  the  curve.  If,  how- 
ever, the  orbit  is  already  known,  and  we  require  to  determine  the 
law  of  force,  the  operation  will  be  easier,  as  no  integration  will  be 
necessary ;  thus,  by  differentiating  (6),  we  have 

These  equations,  or  in  fact  the  single  equation  (5),  contains  the 
whole  theory  of  central  forces,  at  least  as  far  as  regards  the  nature 
of  the  orbit,  the  law  of  the  force,  and  the  velocity  of  the  body  at 
any  point.  When  the  time  enters  into  consideration  the  equations 
(2),  (3),  and  (4)  become  useful. 

(136.)  Having  established  these  equations,  it  would  be  easy  now 
to  deduce  from  tliem  a  variety  of  other  forms,  but  we  shall  not  de- 
tain the  student  by  so  doing.  One  or  two  transformations,  how- 
ever, deserve  to  be  noticed,  on  account  of  their  utility. 

Looking  to  the  terms  within  the  brackets  in  equation  (5),  as  coit- 
nected  with  the  angle  P,  (fig.  106,)  we  observe  that  they  denote 


•  We  must  caution  the  student  against  supposing  that  we  here  depart  from  the 
theory  laid  down  in  the  Differential  Calculus,  by  seeming  to  view  dt  as  a  finite 
quantity  instead  of  absolutely  nothing.  It  must  be  remembered  that  in  every  for- 
mula, containing  only  first  differential  coefficients,  the  independent  variable  may 

be  always  considered  as  entirely  arbitrary ;  that  is,  every  coefficient  such  as  — , 

may  always,  without  at  all  altering  its  value,  be  changed  into  the  more  general 

form  -rj\<  (see  Diff.  Calc.  p.  99.)     In  the  above,  therefore,  we  tacitly  suppose 

this  change  to  be  effected,  and  eliminate,  in  fact,  the  finite  quantity  {dt). 


ON  CENTRAL    FORCES.  169 

{Diff.  Calc.  p.  119)*  -^—-{.1=  .    I 

Now,  if  from  the  pole  a  perpendicular  p  be  demitted  upon  the  tan- 
gent at  P,  its  length  will  be  /J=r  sin./^P  ;  hence  we  may  transform 
the  equation  (5)  into 

an  expression  of  remarkable  simplicity. 

(137.)  Another  useful  and  simple  form  is  obtained  by  introducing 
the  expression  for  the  chord  of  the  osculating  circle  drawn  from  P 
through  the  pole.  This  expression  is  thus  deduced :  from  the 
centre  of  curvature  C  (fig.  107,)  draw  CM  perpendicular  to  the 
radius  vector,  then  PM  will,  obviously,  be  half  the  chord  of  curva- 
ture ;  upon  the  tangent  PT  demit  the  perpendicular  ST:=jo  :  then 
the  angle  TPM  being  equal  to  the  angle  PCM  the  triangles  STP, 
PMC,  are  similar,  therefore, 

chord=2  PM=2^^^-=2y^ (1). 

bP  r  ^  ^ 

Now  if  we  join  SC=a,  and  draw  the  perpendicular  ST'=PT,  then, 
of  whatever  curve  S  is  the  focus  and  CP^y  the  radius  of  curva- 
ture, we  always  have,  (6'eom.  p.  35,)  • 

cLt  (It 

a2=r2 +y2_2y»  .-.  0=2r-i 2y  .-.  2y=2r-j-; 

dp  dp 

dr 
and,  substituting  this  in  (1),  we  have  chord=2p-^  ;  and,  conse- 

quently,  the  equation  (10),  art.   (136),  becomes  by  substitution 

2^2  fly, 

R=— : — r-    Moreover,  since  from  (10),  v^='Rp—=Ry,h.  chord; 
chord  ■'  ^  dp 

and  since  we  likewise  know,  if  the  force  R  were  to  remain  constant 

and  to  draw  a  body  let  fall  from  P  along  the  chord  of  curvature, 

that  when  it  should  have  fallen  through  |  the  chord  we  should  have 

t'^=RX2  chord,  we  infer  that  the  velocity  in  the  curve,  at  any  point, 

is  the  very  same  as  the  body  tvould  acquire  in  falling  through  one 

fourth  the  chord  of  curvature,  supposing  the  force,  at  that  point, 

to  remain  constant. 

(138.)  The  force  F,  which  placed  at  S,  would  compel  the  body 

at  P  to  revolve  round  the  centre,  at  the  same  distance  SP,  with  the 

angular  velocity,  it  actually  has  when  at  P,  is  called  the  centrifugal 

force  at  P,  or  rather  the  centrifugal  force  is  equal  and  opposite  to 


dr 
*  At  the  top  of  the  page  here  referred  to,  the  expressions  r  —  and  —  r2  should 

dh> 
be  interchanged 

P  22 


170  ELEMENTS  OF  DYNAMICS. 

this.     The  expression  for  the  angular  velocity  is,  as  we  have  al- 
ow 
ready  seen,  —  which  is  no  other  than  the  actual  velocity  in  the 

small  circle  which  a  point  in  the  radius  vector  at  the  unit  of  dis- 
tance from  S  describes  as  SP  revolves  ;•  the  velocity,  therefore,  due 

flw 

to  the  force  F,  of  which  we  speak,  is  r~j  ;  but  when  the  body  moves 

in  a  circle  the  centrifugal  force  is  expressed  by  the  square  of  the 
velocity  divided  by  the  radius,  so  that  we  have  here 

F=r-^.-.  F=^  (equa.  3,  art.  135). 

If  the  force  R,  really  placed  at  S,  were  but  just  sufficient  to  retain 
the  body  at  P  in  a  circle,  either  of  these  values  of  F  would  express 
its  intensity;  whatever  additional  influence,  therefore,  R  actually 
exerts,  or  whatever  influence  short  of  this  R  exerts,  it  must  be 
wholly  employed  in  diminishing,  or  increasing  the  radius  vector  SP ; 

this  portion   of  force,  therefore,  is  truly  represented   by  .±-^— ; 

the  upper  sign  applying  when  the  radius  vector  increases,  and  the 
lower  when  it  decreases ;  or  omitting,  as  usual,  the  signs  before 
the  difl^erential  coefficients,  we  have,  for  the  intensity  of  the  central 
or  centripetal  force  R, 

''='-W-lF=^-liF  ■  ■  ■  ■  ^'^-     The  force  ^ 

dr 
is  called  the  paracentric  force,  and  the  velocity  — ,  due  to  it,  the 

paracentric  velocity.  This  velocity  we  may  readily  express  for  any 
point  in  terms  of  the  coordinates,  independently  of  /.  Thus  sub- 
stituting the  value  of — 2/Rrfr,  in  equation  (5),  art.  (135)  in  the 
equation  (4),  which  precedes  it,  and  we  have 

dr"      c"   dr" 

■^=^--^=  (paracentnc  velocity)". 

The  paracentric  force  is  (equa.  1)  the  difference  between  the 
centripetal  and  centrifugal  forces.  ' 

We  shall  now  proceed  to  illustrate  this  theory. 

On  the  Motions  of  the  Planets. 
(139.)  Before  Newton's  discovery  of  the  law  of  universal  at- 

•  For  as  the  angle  w  is  described  in  /",  the  point  to  which  we  allude  describes 
the  arc  1  X  u ;  hence  the  velocity,  being  the  diBerentiai  coefiicient  of  the  space, 

with  respect  to  the  time,  is  — . 
dt 


ON  THE  MOTIONS  OF  THE  PLANETS.  171 

traction  the  paths  in  which  the  planets  revolve  about  the  sun  had 
been  ascertained  by  observation ;  and  the  following  laws,  discovered 
by  Kepler,  and  afterwards  called  Kepler'' s  laivs,  were  known  to  be 
true.    They  are  the  three  following : 

I.  TTie  radius  vector  of  every  planet  describes  about  the  sun  as 
a  pole,  equal  areas  hi  equal  times. 

II.  The  path  of  every  planet  is  an  ellipse,  having  the  sun  in 
one  of  its  foci. 

III.  TTie  squares  of  the  times  of  revolution  are  as  the  cubes 
of  the  mean  distances  from  the  sun,  or  as  the  cubes  of  the  major 
axes  of  the  orbits. 

(140.)  From  these  facts,  revealed  by  observation,  let  us  now  de- 
duce the  law  of  attractive  force,  on  which  they  must  necessarily 
depend.  In  order  that  nothing  may  be  assumed  in  this  inquiry,  let 
us,  before  we  bring  into  application  the  preceding  theory,  show 
that  this  force,  whatever  it  be,  must  be  directed  towards  the  sun. 
In  order  to  this  take  the  centre  of  the  sun  for  the  origin  of  the  rect- 
angular axes,  these  being  in  the  plane  of  the  orbit,  then  the  com- 
ponents of  the  force  acting  on  the  body  at  any  time  <",  in  directions 
parallel  to  these  axes,  will  be 

d^x  d  ^v 

-p^=X,  — -^=Y,  from  which  we  get,  as  at  art.  (122), 

We  have,  moreover,  seen  that  fydx — f-^dy  is  the  double  area 
described  by  the  radius  vector  about  the  sun,  during  the  time  t" 
(122),  and  since  by  Kepler's  first  law,  this  area  is  always  propor- 
tional to  the  time,  we  may  generally  represent  it  by  ct,  c  being 

ydx  —  xdv 
constant ;    hence,  we    have,   by  differentiation, —  =  c ; 

d  .  (vdx  ^—  xdv^ 
and,   differentiating    again,   we   have  —        j- '^  =  0  .  (2) ; 

Y     V 

consequently,  (1)  Xy — Ya?=0  .-.  — =— ;  that  is,  the  lines  repre- 

A.      X 

sented  by  X,  Y,  are  proportional  to  those  represented  by  x,  y, 
(fig.  108),  and,  therefore,  PR  is  in  the  same  line  as  PS,  that  is, 
the  resultant  of  the  forces  on  P  is  in  the  direction  PS,  so  that  the 
planet  P  moves  under  the  influence  of  a  central  force  at  S,  the  place 
of  the  sun.  We  infer  that  this  force  must  be  attractive  and  not  re- 
pulsive, because,  from  the  second  law,  the  curve  PP'  is  always  con- 
cave to  S,  and,  therefore,  P  is  drawn  from  its  wonted  path  Fp  to- 
wards S. 

(141.)  Having  established  this,  we  have  now  only  to  compare 
the  polar  equation  of  the  orbit  with  the  equation  (8),  at  page  168, 


172  ELEMENTS  OF  DYNAMICS. 

in  order  to  dcterniine  the  value  of  R  the  force  of  attraction  on  the 
planet.    In  the  ellipse  referred  to  the  focus,  the  expression  for  r,  is 

(Anal.  Geoin.  p.  165,)  r=—^ —;  in  which  a  is  the  serai-ma- 

^        l-fecos.e 

jor  axis,  OB,  (fig.  110,)  e  the  ratio  of  the  eccentricity  OS  to  the 
semi-major    axis    and  d  the    variable   angle    P8B.     But,    instead 
of  SB,  let  us   take  any  other  fixed  axis,  SX,  making  with  SB  the 
angle  BSX=a,  then   calling  PSX,  w,  g=qj — a,  and,  therefore, 
j.^        a(l—e')         .1^  ^  ^l+fcos.(co— g)  ^  _  ^  ^ 
1  -f  e  cos.  (co — a) '  '  r  a  ( 1 — €'■')         •  •  •  •  v  ;• 

Differentiating  this,  with  respect  to  the  variable  angle  «,  we  have 
du     e  sin.  (« — a)  _ 
d^^ ~~a(l—e'')   ' 

,    I-/..  •     •  .       d^U  eCOS.  (co a)     .^^ 

and  differentiating  again,  -j-t-= 77-^^ — 77-^'  (2)  ;  adding  this  to 

doi^  u  ( 1— C") 

equation  (1)  above,  we  have 

—J—,  +  u  = — — — .    Hence,  by  equation  (8),  page  168 

"~a  (1— e-)  ~  a(l— e*")  "r*  ^  ^ 

The  coefficient  of  — ,  in  this  expression  for  the  central  force  is 

constant  for  the  same  orbit ;  hence  every  planet  is  retained  in  its 
orbit  by  an  attractive  force  residing  in  the  sun,  and  varying 
in  intensity  inversely  as  the  square  of  the  distance  at  which  it 
acts. 

(142.)  For  the  velocity  at  any  point,  we  have 

3  c*  fdr_  2  c''  1 
a{\—c-)J  r^^a(l— O* 
to  determine  the  constant  C  we  must  know,  a.  priori,  the  velocity 
at  some  given  distance  ;  or  we  must  know,  at  what  distance  and 
with  what  velocity  the  planet  is  originally  projected  into  space. 
Calling  this  primitive  velocity  v^,  and  the  corresponding  distance 
fj,  we  have 

2c2      ,  1       1 

hence  the  velocity  is  greatest  when  r  is  least,  that  is,  at  that  extre- 
mity of  the  major  diameter  which  is  nearest  to  the  sun:  this  point 
is  called  the  nearer  apse,  the  curve  being  there  perpendicular  to  the 
radius  vector.  At  the  opposite  point,  or  farther  apse,  the  velocity 
is  least,  since  at  this  point  r  is  greatest. 


^rr^i  3  c*        pdr         2  c''         1       ^ 

■'  ail — e-)J   r^      a  (I — e")   r 


ON  THE  MOTIONS  OF  THE  PLANETS.  173 

It  may  be  remarked  that  the  coefficient  —7- -— ,  which  occurs 

•'  a(l — e^) 

in  the  foregoing  expressions  for  R  and  v,  is  the  value  of  the  at- 
tracting force  at  the  unit  of  distance  from  the  centre,  being  what  R 
becomes  when  r=l;  5C  always  represents  the  space  described  by 
the  radius  vector  in  one  second  of  time. 

(143.)  The  equation  (9)  at  page  169,  will  furnish  a  simpler  ex- 
pression for  the  velociiy  than  that  just  deduced  ;  for  the  perpendi- 
cular p  from  the  focus  on  the  tangent  being  {Anal.  Geom.  p.  139.) 

6V        a^ri — e^)}- 
r,2_ __v J_ 

'        2  a — r         2  a — r 

2_  c^_ c^, 2  a— -r 

'''  ^  ~p~'a2(l— e^'       r  ^'  ^ 

It  will  be  easy  to  compare  this  velocity  with  that  Avhich  a  body 
would  have  revolving  in  a  circle  at  the  same  distance  r,  and  about 
the   same  centre  of  force  ;  for,  calling  this  velocity  v',  we  know 

that  R  = .*.  w'^=Rr ;  hence,  referring  to  the  value  of  R,  in  equa- 

r 

tion  (3)  above,  we  have 

1     •       IT  1      ■  1  3  o  — )'      1 

vel."  m  ellipse  :  vel.^  in  circle  : :  :  —  :  :  2  a — r  :  a  ; 

ar  r 

that  is,  as  the  distance  of  P  from  the  empty  focus  of  the  ellipse  to 

the  semi-major  axis. 

From  the  same  expression  for  v^  we  learn  that  the  velocity  at  the 
mean  distance,  r=:a,  that  is,  at  the  extremity  of  the  minor  axis,  is 
a  mean  proportional  between  the  velocities  at  the  apsides  or  at  the 
distances  r=r',  and  r=2a — r'. 

(144.)  In  the  expression  for  the  force  (3)  the  quantities  a,  e,  c, 
which  enter,  are  different  for  different  planets  ;  we  cannot  conclude, 
therefore,  from  this  expression,  whether,  like  terrestrial  gravity,  this 
force  is  independent  of  the  magnitude  and  constitution  of  the  body 
attracted,  that  is,  whether  at  the  same  distance  r,  or  at  the  unit  of 
distance,  R  has,  in  all  cases,  the  same  value ;  but  Kepler's  third 
law,  which  we  have  not  hitherto  used,  enables  us  to  establish  this 
point.  Let  T"  be  the  time  of  a  complete  revolution  of  any  planet 
P ;  then,  c  being  the  double  area  described  in  1",  cT  will,  by  the 
first  law,  be  the  double  area  described  during  a  whole  revolution, 
that  is,  it  will  express  twice  the  area  of  the  ellipse  ;  but  the  area  of 
this  ellipse  is  (Jnt.  Calc.  p.  123  art.  68)  rf  a^^/  \  1 — e^;  ] 

....T=.,«v!.-e'!.-.^^  =  -45^. 

In  like  manner  with  respect  to  any  other  planet  P' 
p2 


174  ELEMENTS    OF    DYNAMICS. 


a'(l— e'^)  T' 

But,  by  Kepler's  third  law, 

T  :  T'  : :  a"  :  a'~ 


and  each  of  these  expressions  denotes  the  influence  of  the  central 
force  on  each  of  the  planets  P,  P',  at  the  unit  of  distance,  these  in- 
fluences, therefore,  being  the  same,  the  force  is  of  a  similar  nature 
to  that  of  terrestrial  gravity,  influencing  all  bodies  alike  at  the  same 
distance  from  the  centre  of  force. 

(145.)  It  may  here  be  remarked  that  the  same  law  of  force  (equa. 
3)  would  be  established  if  we  did  not  know,  from  observation,  that 
the  paths  of  the  planets  were  ellipses,  but  only  that  tliey  were  conic 
sections  ;  for  the  equation  (3)  would  remain  the  same  except  as  re- 
lates to  the  value  of  e  which  is  either  equal  to,  less  than,  or  greater 
than  1,  according  as  the  curve  is  a  parabola,  an  ellipse,  or  a  hyper- 
bola. Hence  if  a  body  move  in  a  conic  section,  iti  virtue  of  a 
central  force  at  the  focus,  it  must  vary  inversely  as  the  square  of 
the  distance  at  which  it  acts. 

(146.)  Let  us  now  proceed  to  the  inverse  problem,  viz.  to  the 
determination  of  the  orbit  which  a  body  must  necessarily  describe 
about  a  central  force,  which  varies  inversely  as  the  square  of  the 
distance  at  which  it  acts  ;  this  being  the  law  of  force  which,  as  we 
have  before  remarked,  was  discovered  by  Newton  to  be  that  which 
governs  the  planetary  motions. 

Let  R  =— ,  /i  being  the  intensity  of  the  force  at  the  unit  of  dis- 

tance,  or  when  r=l,  then  substituting  this  expression  for  R  in  the 
general  equation  (5),  page  168,  we  have  for  this  particular  law  of 
force  the  following  expression  for  the  velocity,  viz. 

..=£!jl.i!l  +  ,.=?^+C..(l); 

r^^  r''      rfu2^    ^       r  ^  ^  ^ 

the  integral  of  which  is  the  equation  of  the  orbit. 

But,  by  article  143,  equation  (2),  when  the  orbit  is  a  conic  section 
^_  c"     1       dr""  _         c"  2a  — r 

^-75l7i--^+M-„,(i_,a)-  —;.         (2;; 

the  integral  of  this  equation  is,  therefore,  the  equation  (1)  p.  172, 
involving  the  arbitrary  constant  o,  and  this,  be  it  observed,  is  true 
whatever  be  the  values  of  the  constants  a  and  e.  But  if  these  be  de- 
termined so  that 

(^'•••"=M.-W''-'=- -«••'">' 

the  two  equations  (1),  (2)  will,  obviously,  become  identical;  hence 


MOTIONS    OF    THE    PLANETS.  175 

with  these  conditions  equation  (1)  will  also  be  the  integral  of  (1) 
above,  and,  in  which  c  and  a  are  fixed  by  the  equations  (3)  and  (4)  : 
(1),  therefore,  is  the  equation  of  the  orbit  sought. 

We  are  warranted,  therefore,  in  inferring  the  converse  of  the  pro- 
position at  the  close  of  last  article,  viz.  that  if  the  central  force  vary 
inversely,  as  the  square  of  the  distance,  the  body  must  describe  a 
conic  section,  having  that  force  in  its  focus. 

(147.)  Having  thus  determined  the  nature  of  the  orbit,  let  us  en- 
deavour to  ascertain  its  form,  which  will  require  the  determination 
of  the  constants  c  and  C,  both  of  which  depend  upon  the  circum- 
stances of  the  initial  motion  of  the  body. 

C  may  be  determined  from  knowing  the  initial  velocity  and  dis- 
tance, that  is,  the  impulsion  with  which  the  body  is  launched  into 
space,  and  the  distance  of  the  point  of  projection  from  the  centre  of 
force,  call  these  respectively  v^  and  r^,  then  from  equation  (1) 

C=V-?^..(l). 

To  determine  c  requires  that  we  know  not  only  the  point  and  velo- 
city of  projection,  but  also  its  direction  ;  or  the  angle  it  makes  with 
the  radius  vector  at  that  point.  Call  this  angle  6,  then  the  initial 
velocity,  in  the  direction  perpendicular  to  the  radius  vector,  is  Uisin.0, 

which  is,    therefore,    equal"  to  ''it-;  but  equation  (3),  p.  167, 

„    dat  , 

fi^   — =c  ;  consequently, 

c=.r^  Vi  sin.  9  .  .  .  .  (2). 
Hence  the  constants  which  enter  into  the  equations  (3)  and  (4) 
are   determined   in    terms    of  the    initial  quantities ;    substituting 

them  in  those  equations,  we  get,  from  (4),  a  = i~^lTT  '     ^'^^' 

from  (3),  e=  ^1  -  -  =^1  +  j^- {v,--  -). 

By  means  of  these  two  equations  the  orbit  may  be  constructed  ; 
its  form  may  be  completely  determined  by  the  second  equation  alone, 
since  the  form  depends  entirely  upon  the  value  of  e.  It  will  be  an 
ellipse  if  e<;l,  an  hyperbola  if  ei>\,  and  a  parabola  if  e=l ;  that 

is,  the  orbit  will  be  an  ellipse  if  Uj  ^  <C — ;  an  hyperbola  if  u^  ^>> — ; 

a  parabola  if  v  ^  ^  =  —  ; 

so  that  the  same  central  force  which  governs  the  planetary  motions, 
and  causes  them  to  describe  ellipses,  would  have  been  equally  com- 
petent to  cause  them  to  describe  either  hyperbolas  or  parabolas,  but 


176  ELEMENTS    OF    DYNAMICS. 

not  any  other  curves.  For  aught  we  know  to  the  contrary,  there- 
fore, there  in:iy  be  planets  governed  by  the  same  attractive  influence, 
and  moving  in  hyperbolic  or  parabolic  orbits,  and  which,  therefore, 
continually  proceed  onward  in  space  without  ever  returning. 

It  is  a  singular  fact,  that,  as  the  foregoing  expression  for  a,  the 
semi-major  axis  of  the  orbit,  is  independent  of  d,  the  angle  of  pro- 
jection, the  major  axis  of  the  orbit,  will  be  of  the  same  length  what- 
ever be  the  angle  of  projection  ;  the  minor  axis,  however,  will  vary 
with  this  angle,  seeing  that  its  sine  enters  into  the  expression  for  the 
eccentricity. 

As  to  the  absolute  lengths  of  these  axes,  they  are  at  once  deter- 
mined when  the  initial  conditions  of  the  motion  are  given  by  means 
of  the  equations  (3)  and  (4),  since  the  constants  which  enter  them 
are  known  in  terms  of  these  given  conditions  by  equations  (1)  and 

(2)- 

The  velocity  of  the  body  at  any  point  of  its  orbit  is  given  by 

the  equation  (2)  art.  (146),  and,  by  substituting  this  expression, 
for  V  in  the  general  equation  (5),  art.  (135),  we  readily  get  a  dif- 
ferential expression  for  the  angle  to  corresponding  to  that  point,  and, 
finally,  the  value  of  du,,  given  by  this  expression,  substituted  in  the 
equation  (3)  of  the  same  article,  will  furnish  a  differential  equa- 
tion between  t  and  r,  and  thence  the  time  of  arriving  at  the  pro- 
posed point  becomes  known. 

In  all  these  results  the  quantity  h  enters,  which  quantity  denotes 
the  value  of  the  accelerative  force  at  the  unit  of  distance  from  the  cen- 
tre of  attraction  ;  or  rather  it  expresses  the  number  of  these  units 
in  the  linear  space  which  measures  the  attractive  force  at  the  unit 
of  distance.  Now  the  attractive  forces  of  the  sun  and  planets,  corres- 
ponding to  any  proposed  distance,  vary  directly  as  their  masses,  there- 
fore, whatever  energy  the  unit  of  mass  exerts  at  the  unit  of  distance 
the  mass  M  of  the  sun  exerts  M  times  as  much,  and  the  mass 
m  of  the  planet,  m  times  as  much ;  the  whole  force,  therefore, 
which  the  sun  regarded  as  fixed,  exerts  on  the  planet  at  the  unit 
of  distance,  is  (art.  119)  M-|-?)i  times  as  much  as  that  exerted  by 
the  unit  of  mass  at  the  unit  of  distance.  Considering  this  latter  to 
be  the  unit  of  force,  and  representing  it  by  1,  accordingly  M+m 
will  truly  represent  the  attractive  power  exerted  on  the  planet  at 
the  unit  of  distance,  and,  therefore,  at  r  such  units  distant  tlie  ex- 

pression  for  the  attractive  force  will  be — ,  which  is  the  value 

h  . 
of  —in  the  preceding  formulas  when  we  consider  the  action  of  the 

sun  on  a  single  planet  only.  We  may  also  express  the  intensity 
of  the  solar  and  planetary  attractions   in   terms  of  terrestrial  gra- 


ON    THE   MOTION    OF    A    SOLID    BODY.  177 

vity  ;  thus,  calling  the  radius  of  the  earth  r^,  and  m^  its  mass,  we 
have,  since  the  attractions  are  directly  as  the  masses  and  inversely 
as  the  squares  of  the  distances, 

m,    M           Mr,'' 
—  :  —:  :  sr : sr, 

the  attractive  force  of  the  sun  at  any  distance  r ;  and,  in  like  man- 
ner,  the  attractive  force  of  the  planet  is  — —„^ ;    hence  their  united 

influence  is ^ .  g,  the  factor  which  multiplies  g  being  an 

abstract  number.  This  number  will,  of  course,  remain  the  same  if 
we  deliver  the  mass  and  distance  each  from  its  peculiar  unit,  re- 
garding M,  wi,  r^,  &c.  to  be  numbers  merely,  in  which  case  writing 

M+m     r,^  \      r/      .      , 

the  expression  thus .  -^—k\  we  see  that  —^2*  is  that  which 

we  have  taken  above  for  the  unit  of  attractive  force  ;  it  is  plainly 
the  expression  for  the  attraction  of  the  mass  1  at  the  distance  1 . 

We  here  close  the  second  section ;  for,  to  pursue  these  interest- 
ing inquiries  further,  we  should  be  compelled  to  pass  by  matters 
more  especially  entitled  to  a  place  in  a  treatise  on  Mechanics.  We 
have,  however,  given  thus  much  of  the  first  principles  of  Physical 
Astronomy,  because  we  Avere  unwilling  altogether  to  omit  touching 
upon  a  subject  so  highly  calculated  to  excite  the  inquiries  of  the 
student,  and  because,  moreover,  we  had  indulged  a  hope  of  being 
able  to  unfold  these  first  principles  with  more  simplicity  than  is 
usually  done. 


SECTION  III. 
ON  THE  MOTION  OF  A  SOLID  BODY. 

(148.)  Hitherto  we  have  limited  our  investigations  almost 
solely  to  the  motions  of  a  single  point :  or,  if  we  have  at  any  time 
introduced  the  consideration  of  a  moving  body,  it  has  usually  been 
on  the  hypothesis,  that  the  influence  by  which  it  moved  was  dif- 
fused uniformly  through  its  mass,  acting  alike  upon  every  particle ; 
so  that  if  any  portion  of  the  body  were  to  be  removed,  all  the  in- 
fluence, in  virtue  of  which  that  particular  portion  moved,  would  be 
taken  away  too,  and  the  remaining  part  of  the  mass  would  go  on 
precisely  in  the  same  way  as  it  would  have  done  if  accompanied  by 
the  part  subtracted,  and  precisely  as  it  would  do  if  reduced  to  a  sin- 

23 


178  ELEMENTS  OF  DYNAMICS. 

gle  particle.  It  is  in  this  way  that  the  forces  of  attraction  are  uni- 
formly difTused  through  the  masses  of  the  attracted  bodies,  and  to 
this  description  of  forces  our  attention  has  been  almost  entirely 
directed  hitherto.  We  have  not,  however,  wholly  overlooked 
those  motions  which  result  from  pressure  constantly  pushing  for- 
ward a  mass  of  matter,  not  tliat  we  mean  to  mark  any  difference, 
as  far  as  effects  are  concerned,  between  pressure  dynamically  con- 
sidered, and  an  attractive  force  ;  for,  as  already  observed  at  (104), 
the  motion  produced  by  an  attractive  force  may  be  considered  as 
due  to  the  body's  own  pressure,  as  is  manifest ;  but  if  it  were  to 
be  urged  by  a  pressure  less  than  this,  we  ought  obviously  to  ex- 
pect a  diminished  acceleration ;  and  if  it  were  urged  by  a  greater 
pressure,  we  should  look  for  an  increased  acceleration  ;  but,  as  just 
remarked,  particular  cases  of  this  kind  have  already  been  consider- 
ed, viz.  in  problems  I.  and  II.  at  page  129  ;  thus  referring  to  the 
first  of  these  problems,  we  find  that  the  whole  weight  to  which 
motion  is  given,  is  VV-fWj,  which  represents  the  whole  pressure 
of  the  mass  moved ;  but  the  motion  is  due  to  a  less  pressure,  viz. 
to  Wj  sin.  I,  —  W  sin.  '^  i,  and  accordingly  we  find  a  diminished 
acceleration. 

(149.)  The  pressure  which  thus  moves  a  body  is  called  a  moving 
pressure,  or  rather  a  moving  force  ;  the  acceleration  due  to  such  a 
ibrce,  or  the  force  competent  to  produce  any  proposed  acceleration, 
may  be  determined  by  the  principle  at  page  121  :  thus,  calling  the 
whole  pressure  or  weight  of  any  mass  ^I,  W,  the  acceleration  due 
to  this  pressure  g,  and  that  produced  by  any  other  pressure  or 

F 
moving  force,  F,  we  have  o' :  F  :  :  W  :  W  — =the  moving  force, 

o 

which  expression  obviously  represents  the  weight  which  the  mass 
M  would  have  if  the  accelerative  force  which  impressed  that  weight 
were  F. 

The  weights  of  bodies  obviously  vary  with  their  mass,  or  with 
the  quantity  of  matter  they  contain,  and  also,  as  the  accelerative 
force  which  impresses  weight;  that  is  to  say,  weight  varies  con- 
jointly as  the  mass  and  the  accelerative  force ;  we  may,  therefore, 
in  these  inquiries  substitute  for  any  weight  W  the  quantity  to 
which  it  is  always  proportional,  viz.  the  quantity  Mg";  M  being  the 
mass  or  rather  the  number  of  times  the  body  contains  some  fixed 
unit  of  mass,  and  "■  being  the  value  of  gravity,  or  the  force  which 
impresses  weight  on  the  mass.  Representing  the  weight  W  by 
M^",  and  putting  ^  for  the  moving  force,  we  have,  by  the  foregoing 
expression, 

*=MF=M -j-=M-7— , 
^  dt  dt* 


ON    THE    COLLISION    OF    BODIES.  179 

sijpposing,  as  we  here  do,  that  each  particle  of  the  moving  mass 
Jias  a  common  velocity  at  every  instant. 

As  it  immediately  follows  from  this  that  F=—  ,  we  see  that  the 

acceleration  is  as  the  moving  force  or  pressure  directly,  and  as  the 
mass  inversely. 

For  the  velocity  at  any  time  t",  we  have,  supposing  ^  constant, 

dv^—  (It  .•.  v=  :^  t,  so  that  the  velocity  acquired  in  any  time  i", 

is   the  same,  so  long  as  the  ratio,  —  ,  of  the  moving  force  to  the 

mass  moved  is  the  same. 

It  is  usual  to  call  the  product  My  of  the  moving  mass  by  its  ve- 
locity the  momentinn,  so  that  as  accelerative  force  is  represented  by 
the  differential  coefficient  of  the  velocity  relatively  to  the  indepen- 
dent variable  /,  the  moving  force  is  represented  by  the  differential 
coefficient  of  the  momentum  relatively  to  t.  Instead  of  mass,  how- 
ever, we  may,  if  we  please,  always  substitute  weight,  since,  as  re- 
marked above,  these  are  always  proportional. 

We  shall  terminate  these  introductory  remarks  by  observing,  that 
all  the  deductions  at  art.  105,  respecting  the  acceleration,  velocity, 
space,  and  time,  apply  equally  here  ;  and  all  the  formulas  em- 
ployed there  become  suited  to  the  present  inquiry  M'hen  -rj-  is  sub- 
stituted in  them  for  F  ;  provided,  as  before  mentioned,  that  the  mov- 
ing force  impresses  a  common  velocity  on  all  the  particles  of  the 
mass  moved,  so  that  the  acceleration  of  the  whole  may  be  that  of 
each  particle. 


CHAPTER  I. 

ON    THE    COLLISION  OF    BODIES. 

(150.)  In  the  present  chapter  we  propose  briefly  to  consider 
the  circumstances  of  the  motions  of  bodies  moving  in  certain  di- 
rections, from  the  effects  of  impulsion  and  impinging  against  each 
other. 

Let  us  suppose  that  the  quantity  of  matter  which  we  assume  for 
the  unit  is  projected  by  any  given  impulsion ;  it  will  move  with  a 
constant  velocity  always  proportional  to  the  intensity  of  the  force 
of  impulsion  (p.   118);  this   velocity,  therefore,  will  always  cor- 


180 


ELEMENTS    OF    Dl-XAMICS. 


rectly  rcprpsent  tho  intensity  of  the  force.  If  we  take  two  such 
units,  and  two  such  imj)ulsions  act  on  them  simultaneously,  and  in 
the  same  direction,  the  relative  positions  of  tlic  moving  masses  will 
be  always  preserved,  so  that  if  the  two  units  were  actually  blended 
into  one  mass,  and  the  two  impulsions  be  thus  made  to  coalesce  and 
form  a  double  impulsion,  the  same  velocity  as  before  would  be  im- 
pressed on  the  mass  ;  and  it  is  plain  that  in  like  manner  if  M  such 
units  be  blended  into  one  mass,  which  receives  an  impulsion  of  M 
times  the  intensity  we  have  supposed  applied  to  the  unit,  still  the 
same  velocity  would  be  impressed  ;  consequently,  when  any  body 
whose  mass  is  M  moves  from  the  effect  of  an  impulsive  force,  the 
correct  expression  for  the  intensity  of  that  force  must  be  My,  tjie 
mass  into  the  velocity,  that  is  the  momentum,  measures  the  impulsive 
force. 

Knowing  then  how  to  estimate  the  intensity  of  impulsive  force, 
we  may  proceed  to  consider  the  circumstances  of 

Direct  Impact. 

(151.)  We  shall  first  consider  the  bodies  which  impinge  to  be 
entirely  inelastic,  or  of  such  a  nature  that  they  are  blended  by  the 
impact  into  one  mass,  and  our  object  will  be  from  knowing  the 
forces  of  the  impinging  bodies  to  determine  the  motion  of  the  united 
mass. 

Problem  I. — Two  inelastic  bodies  M,  M,  move  in  the  same 
straight  line  with  velocities  t",  v^  :  to  determine  the  velocity  after 
impact. 

The  impulsive  force  on  M  is  Mr  ;  that  on  Mj  is  M,Vj,  and  these 
two  combined  form  the  impulsive  force  \»hich  moves  the  blended 
mass  M  +  Mj  ;  but  if  V  be  the  velocity  of  this  mass,  the  impulsive 
force  on  it  must  be  (M  +  MJ  V, 

.'.  (M+MJ  V=Mt;+M,f,  .-.  V -±^^  . .  (1) 

the  velocity  required.  If  the  body  M^  were  moving  in  a  direction  op- 
posite to  M  so  as  to  meet  it,  then  v^  should  be  taken  negatively,  and 

in  that  case,  V= — ,,    \i— ^  .    (2),  and  the  sign  of  V  thus   deter- 
M  +  M,  ^  ^  ^ 

mined  will  point  out  the  direction  in  which  the  mass  moves  after 

impact. 

Mu 
If  the  body  M,  be  at  rest,  then,  since  v,  =0,  V^^^ — r-r  .  (3). 
•^      *  '  1       '         M  +  M,    ^  ^ 

In  all  cases  the  momentum  (M  +  MJ  V  after  impact  is  the  sum 

of  the  momenta  before  impact,  those  being  taken  with  opposite 

signs  which  act  in  opposite  directions ;  so  that  the  momentum  lost 


ON   THE    COLLISION   OF    BODIES.  181 

by  the  one  body  by  the  collision,  is  precisely  that  which  is  gained 
by  the  other.  In  the  last  of  the  above  cases  (3),  the  momentum  of 
the  whole  mass  being  (M-fMj)  \  =  Mv,  and  the  momentum  gained 
by  M,  which  was  at  rest  being  M^Vj  this  must  be  the  momentum 
lo'st  by  M. 

This  loss  of  force  in  M  shows  that  the  mass  M^  opposes  a  re- 
sistance to  the  communication  of  motion,  and  M^V  expresses  the 
value  of  that  resistance,  or  the  impulsive  force  necessary  to  balance 
it ;  this  result  is  quite  independent  of  the  weight  of  the  resisting 
body,  seeing  that  we  here  consider  only  its  mass  ;  it  is,  therefore, 
the  consequence  of  its  inertia,  which  is  therefore  proportional  to 
the  mass. 

(152.)  Let  us  now  consider  the  impact  of  two  elastic  bodies 
which,  as  in  the  former  case,  move  so  as  to  impinge  at  some  point 
in  the  common  line  described  by  their  centres  of  gi'avity. 

Bodies  of  this  class  yield  to  the  force  of  impact,  and  sutler  a  com- 
pression, and  therefore  a  change  of  figure  ;  and  the  elasticity  is  that 
inherent  force  Avhich  the  body  exerts  to  recover  its  original  form. 
If  the  force  thus  exerted  at  every  point  throughout  the  whole  depth 
of  the  impression,  while  the  body  is  recovering  its  form,  is  equal  to 
the  impressing  force  at  that  point,  the  oi'iginal  form  of  the  body  must 
be  perfectly  restored,  and  the  elasticity  is  then  said  to  be  perfect.* 
In  this  case,  whatever  velocity  one  body  lost  during  the  action  of 
the  force  of  compression,  it  afterwards  lost  just  as  much  more  during 
the  action  of  the  force  of  restitution  ;  and  whatever  velocity  the  other 
body  gained  during  the  compression,  it  gained  as  much  more  during 
the  restitution ;  for  the  compressing  and  restoring  forces  (which 
are  no  other  than  continued  pressures,  although  operating  for  an 
exceedingly  short  time)  are  equal,  and  therefore  equally  oppose  the 
motion  of  one  of  the  bodies,  and  equally  favour  the  motion  of  the  other. 

It  is  easily  seen  that  the  same  would  be  true  if  only  one  of  the 
bodies  were  perfectly  elastic,  and  the  other  perfectly  hard,  or  inca- 
pable of  receiving  an  impression. 

When  the  force  of  restitution  is  not  equal  to  that  of  compression, 
but  only  the  eth  part  of  it,  the  elasticity  is  said  to  be  imperfect,  and 
e  measures  its  relative  intensity ;  it  is  competent  to  communicate 
only  the  eth  part  of  the  velocity  due  to  perfect  elasticity. 

Problem  II. — Two  elastic  bodies  M,  M^,  moving  with  velocities 
V,  v^,  strike  with  direct  impact:  to  determine  their  velocities  after- 
wards. 

So  long  as  the  compression  continues,  the  bodies  move  as  one 

*  The  restitution  is  moreover  considered  to  occupy  the  same  length  of  time  as 
the  compression. 

o 


182  ELEMENTS  OF  DYNAMICS. 

mass,  and  therefore  with  the  velocity  V  =  — rr— ^t^-^s  so  that  M  will 

lose  the  velocity  v  —  V  and  M,  will  lose  the  velocity  v^  — V,  and 
these  would  express  the  velocities  communicated,  but  in  opposite 
directions,  by  the  force  of  restitution,  if  the  elasticity  were  perfect; 
but  let  it  be  only  the  fth  part  of  perfect,  e  being  a  fraction,  then  the 
additional  velocity  lost  by  M  will  be  e  (v  —  V),  and  byM,, 
e(v^ — V);  hence  the  velocities  after  the  impact  will  be 

of  the  body  M,    V  =v  —  (1-f  e)  (v  —  V) (1); 

of  the  body  M,,  V"=u,— (1+e)  (v^—Y) (2); 

or,  substituting  for  V  its  value  above, 

V=..-(l+e)4^-^)  ....  (4). 

If  c=0,  that  is,  if  the  bodies  are  inelastic,  then,  as  is  plain  from  the 
equations  (1)  (2),  the  velocities  after  impact  are  V'  =  V,  V"=V,  as 
we  know  they  ought  to  be.  If  e=l,  that  is,  if  the  elasticity  is  per- 
fect, then,  from  equations  (3)  and  (4), 

If  we  multiply  each  mass  by  the  velocity  of  it  after  impact,  we  have, 
when  the  bodies  are  perfectly  elastic, 

MV'  +  MiV"=Mu+M,  v^ (7) ; 

hence  the  sum  of  the  momenta  before  impact  is  the  same  as  the 
sum  after  impact. 

If  we  subtract  the  equation  (2)  from  (1),  taking  e=l,  we  have 
\'—Y"=v  —  2v  —  v^-\-2v^=v^—v  ....  (8); 
hence  the  difference  of  the  velocities  is  the  same  both  before  and 
after  impact. 

When  the  bodies  are  perfectly  equal,  as  well  as  perfecdy  elastic, 
they  exchange  their  velocities,  each  moving  after  the  impact  with 
the  velocity  the  other  had  before  the  impact ;  this  follows  from  put- 
ting M=Mi,  in  the  equations  (5),  (6) ;  if,  therefore,  one  body  be 
at  rest,  the  other,  which  strikes  it,  will  impart  to  it  its  entire  velo- 
city, and  rest  in  its  place. 

From  the  equations  (7)  and  (8)  we  have  M  (V'^-i')=M, 
(Vj — V");  V'-|-t'  =  i",+V"  ;  multiplying  these  together  there  re- 
sults MCV'"  — v2)  =  M,  (I'l"  — V"'')or 

MV'^  +  M,  Y"-=Mv'+M,v,^ (9) ; 

that  is,  the  sum  of  the  products  of  each  body  into  the  square  of  its 
velocity  is  tlie  same,  both  before  and  after  impact.     The  mass  into 


ON   THE    COLLISION    OF    BODIES.  183 

the  square  of  the  velocity  is  called  the  vis  viva,  or  living  force; 
SO  that  in  the  collision  of  perfectly  elastic  bodies  there  is  no  loss 
of  vis  viva. 

If  one  of  the  bodies  is  an  immoveable  mass  we  may  consider  it 
in  the  light  of  a  mass  M  j  at  rest,  and  infinitely  large  ;  on  which 
supposition  equation  (3)  gives  for  the  velocity  of  M  after  impact 
V'=  —  ev  ....  (10);  that  is,  as  we  might  expect,  M  would  re- 
bound with  the  same  velocity  with  which  it  struck  the  immoveable 
mass.  It  is  true  that,  conformably  to  our  hypothesis,  the  immove- 
able mass  is  supposed  to  be  perfectly  elastic  as  well  as  the  imping- 
ing body ;  but  the  motion  would  be  the  same  if  it  were  perfectly 
hard,  because  the  force  of  restitution  would  still  be  the  same  as  that 
of  compression,  and  the  whole  of  this  would  be  exerted  to  repel  the 
body. 

On  Oblique  Impact. 

(153.)  When  the  centres  of  gravity  of  two  impinging  bodies  do 
not  move  in  the  same  straight  line,  yet  if  at  the  instant  of  collision, 
the  shock  which  each  receives  be  directed  towards  its  centre  of 
gravity,  the  effects  will  be  calculable  by  the  preceding  formulas. 
Thus  suppose  two  spherical  bodies  M,  Mj,  (fig.  Ill,)  move  so  that 
their  centres  describe  the  lines  PA,  QB,  and  that  they  strike  each 
other  at  the  point  C  ;  we  may  decompose  each  of  the  velocities 
with  which  the  bodies  impinge  into  two,  one  in  the  direction  of  a 
common  tangent  to  the  bodies  at  C,  and  the  other  perpendicular  to 
this  tangent ;  the  components  perpendicular  to  the  tangent  will  be 
those  to  which  the  impact  is  entirely  due,  the  other  components  will 
merely  express  the  lateral  velocity,  or  that  parallel  to  the  tangent 
plane,  which  the  respective  bodies  had  before  impact,  and  which, 
as  nothing  opposes  it,  they  must  still  retain.  Hence,  having  de- 
termined the  velocities  consequent  upon  the  direct  impact  due  to  the 
effective  components  of  which  we  have  spoken,  by  means  of  the 
formulas  in  the  preceding  problems,  we  shall  then  have  to  compound 
these  velocities,  each  with  what  the  body  originally  had  in  the  di- 
rection parallel  to  the  tangent  plane  at  their  point  of  concourse. 

As  an  example,  let  us  take  the  case  where  one  of  the  spheres  is 
at  rest  and  infinite  in  magnitude,  or,  which  is  the  same  thing,  let 
the  body  struck  be  an  immoveable  plane  :  let  the  velocity  given  to 
the  impinging  body  be  v,  the  angle  its  direction  makes  with  the 
plane,  that  is,  the  angle  of  incidence  a,  then  the  components  of  the 
velocity  are  v  cos.  a,  v  sin.  a;  the  latter  of  these  being  perpendicu- 
lar to  the  plane  produces  the  impact ;  therefore,'  if  e  measure  the 
elasticity  of  the  body,  the  velocity,  after  the  impact,  will  be  (equa. 
10)  Y'=:ev  sin.  a,  and  the  direction  will  be  perpendicular  to  the 
plane  ;  hence  the  velocity,  after  impact,  will  be  the  resultant  of  the 


184  ELEMENTS  OF  DYNAMICS. 

velocities  v  cos.  a,  and  e  v  sin.  a,  of  which  the  directions  are  per- 
pendicular to  each  otiicr ;  therefore  the  velocity  of  reflection  will 
be  Vx/{cos.^a-\-€^  sin. -a J  ;  and  the  angle  a'  of  reflection  will  be 

ev  sin.  a 

tan.  a  = =e  tan.  a ; 

V  COS.  a 

so  that  if  the  elasticity  be  perfect,  the  velocity  and  the  angle  of  re- 
flection will  be  respectively  equal  to  the  velocity  and  the  angle  of 
incidence. 

For  a  more  enlarged  view  of  the  theory  of  percussion  and  impact 
the  student  may  consult  Dr.  Gregory's  Mechanics,  vol.  i.  and  the 
Mecanique  of  Poisson,  tom.  ii. ;  and  for  a  variety  of  examples  re- 
ference may  be  made  to  Bridge'' s  Mechanics,  p.  150  et  seq.  We 
omit  practical  examples  here,  which,  however,  are  very  easily 
framed,  in  order  to  make  room  for  more  important  matter. 


CHAPTER  II. 

THE  PRINCIPLE  OF  d'aLEMBERT. 

(154.)  Inquiries  concerning  moving  forces  may  be  sometimes 
considerably  facilitated  by  the  aid  of  a  very  simple  and  very  gene- 
ral proposition,  first  introduced  into  Dynamics  by  D\^lcmbert:  it 
may  be  regarded  as  a  dynamical  axiom,  and  may  be  announced  as 
follows  : 

If  there  be  any  system  of  bodies  A,  B,  C,  &;c.  which  in  virtue 
of  the  forces  applied  to  them  would,  if  entirely  free,  receive  the 
several  velocities  a,  h,  c,  Sic.  but  which,  on  account  of  their  mutual 
connexion,  receive  instead  the  velocities  o,  ji,  y,  Sic.  then  it  is  evi- 
dent that  if  the  velocity  a,  impressed  on  A  were  resolved  into  two, 
of  which  one  is  the  velocity  a,  actually  received,  and  the  other  some 
velocity  a' ;  and  if,  in  like  manner,  the  velocity  b  impressed  on  B 
were  resolved  into  two,  viz.  the  actual  velocity  j3,  and  some  other 
/3' ;  and  if  a  similar  decomposition  be  effected  for  each  impressed 
velocity,  the  forces  due  to  the  component  velocities  a',  ji',  y'.  Sic. 
if  severally  applied  to  the  bodies  A,  B,  C,  &c.  of  the  connected 
system  would  keep  the  system  in  equilibrium. 

For  by  the  decomposition,  which  has  been  effected,  all  the  mo- 
tion which  actually  has  place  must  be  due  to  the  other  components, 
and,  therefore,  thoise  of  which  we  speak  must  destroy  themselves  in 
consequence  of  mutual  actions  of  A,  B,  C,  Sic.  on  each  other. 

It  follows,  therefore,  that  there  must  always  be  an  equilibrium 
between  the  impressed  forces  and  the  actual,  or  effective,  forces, 


THE    PRINCIPLES    OF    D  ALEMBERT.  185 

these  latter  being  taken  opposite  to  their  real  directions,  that  is  to 
say,  if  to  the  body  A  we  apply  the  force  originally  impressed,  and 
also  the  force  due  to  the  actual  motion  of  A,  this  latter  being  oppo- 
site to  its  real  direction ;  and  if  we  do  the  same  with  all  the  other 
bodies  B,  C,  &c.  the  whole  system  will  be  kept  in  equilibrium. 
For  although  the  effective  force  on  either  of  the  bodies  is  not  equi- 
valent to  the  impressed  force,  yet,  as  we  have  just  seen,  the  im- 
pressed forces  may  be  considered  as  resulting  from  the  effective 
forces,  combined  with  another  set  which  destroy  each  other;  hence 
as  these  equilibrate,  the  system  must  equilibrate  by  the  combined 
action  of  the  impressed  forces  with  the  effective  forces  taken  in  op- 
posite directions. 

The  forces  here  spoken  of  are,  of  course,  moving  forces  or  sim- 
ply momenta;  that  is,  continued  pressures,  or  pressures  of  but 
momentary  duration. 

As  an  illustration  of  the  foregoing  general  principle  we  may  take 
the  problem  already  solved,  at  page  129,  viz.  to  determine  the  mo- 
tions of  two  weights  W,  W^,  along  inclined  planes,  placed  back  to 
back,  the  weights  being  connected  by  a  thread. 

Let  us  first  ascertain  the  impressed  forces,  or  those  in  virtue  of 
which  the  bodies  would  move  if  unconnected,  these  are  evidently 
for  W,  W  sin.  i,  and  for  W^,  W^  sin.  i.  Let  us  now  determine 
the  effective  or  actual  forces  ;  for  this  purpose  call  the  velocity  of 
W,  V ;  and  that  of  W^,  v^ ;  then  as  we  know  (149)  that  the  mo- 
tive force  is  always  equal  to  the  mass  multiplied  by  the  accelera- 

W    dv*  W*     dv 

tion,  the  effective  forces  are  on  W, -^  and  on  W,, ~. 

g     dt  g      dt 

Now,  by  the  foregoing  principle,  if  these  forces  taken  with  con- 
trary signs,  that  is,  taken  negatively,  be  simultaneously  applied  to 
the  respective  bodies  with  the  former  forces,  the  system  will  be  in 
equilibrium ;  that  is,  there  will  be  an  equilibrium  if  to  the  two 
bodies  W,  Wj  at  rest,  there  be  applied  the  respective  forces 

w    •      .    ,     W     rfi;       ,  „r     .      .         W     dv^ 

W  sin.  t.  A •  —r  and  W  ,  sm.  % , .  — -; 

g      dt  ^  ^         g       dt' 

therefore,  as  these  must  pull  the  thread  in  opposite  directions,  we 

must  have  the  equation 

.'      W     rfu  .      .        Wj     dv^ 

W  sm.  I  H --  =  W  1  sin.  i, •  —-•. 

g      dt         ^  1         g        dt' 

*  Regard  must,  of  course,  be  paid  to  the  signs  of  the  acting  forces ;  so  that 
if  we  consider  a  force  applied  to  a  body  in  one  direction  to  be  positive,  we  must 
consider  any  other  force  applied  in  an  opposite  direction  to  be  negative ;  there- 
fore, as  we  here  suppose  the  effective  force  on  W  to  pull  it  up  the  plane,  we 
must  tak^t  negatively,  because  we  have  taken  the  impressed  force  on  it,  which 
would  pull  it  do-wn  positively. 

q2  24 


186  ELEMENTS  OF  DYN.tMICS. 

therefore,  since  v  is  necessarily  equal  to  t^,  we  have 
5  _^_|.  Zj!  ^  =  W,  sin.  L  — W  sin.  i 
dv       W.  sin.  i.  —  W  sin.  i 


"  dt  W  +  W,  ^  ' 

•which  is  the  value  of  the  accelerative  force. 

This  solution  it  may  be  observed  is  not  so  simple  as  that  ^ven 
upon  different  principles  at  page  129  ;  but  it  may  serve  to  illus- 
trate D'Alembert's  principle.  It  may  not,  however,  be  amiss  to  re- 
mark here  that  the  solution  of  this  and  of  other  similar  problems 
may  be  readily  obtained  by  equating  two  different  expressions  for 
the  effective  moving  force.  Thus  in  the  present  problem  the  effective 
moving  force  is  obviously  Wj  sin.  i^ — W  sin.  i  ;  it  is  also 
W      dv      W,       dv^ 

and,  therefore,  we  have  the  same  equation  as  above. 

We  shall  give  another  illustration  of  D'Alembert's  principle  in 
this  place. 

Two  weights  Wj,  W,  are  attached,  the  one  to  a  wheel,  and  the 
other  to  its  axle,  to  determine  their  motions. 

The  impressed  forces,  or  those  in  virtue  of  which   the  bodies 

would  move,  if  free,  are  the  weights  themselves ;  the  effective  forces, 

or  those  in  virtue  of  which  they  actually  move,  are  as  in  the  last 

problem 

W    dv  W     dv 

•  -7-  on  W,  supposing  this  to  ascend,  and  — -  •  -r 

g     dt  '      il^        »  g       dt 

on  Wj,  hence  the  system  will  be  in  equilibrium  when  to  the  bo- 
dies W  and  Wj  there  are  applied  the  respective  forces 

„,      W     dv       ,„,  W,     dv. 

W+  —  .  —  and  W, --ir'^ 

g      dt  ^         g        dt 

and  as  these  acting  at  the  extremities  of  the  radii  r  and  R,  of  the 
axle  and  of  the  wheel,  tend  to  turn  the  system  in  opposite  direc- 
tions about  the  axis,  we  must  have  the  equation 

W     dv  W        dv 

,(VV+^.|)  =  R(W,-^.i^)....(l); 

the  relation  between  v  and  v^  is  easily  determined,  for,  as  the  wheel 
must  turn  in  the  same  time  as  the  axle,  the  velocities  of  the 
weights  attached  to  them  must  be  as  their  radii, 

R         dv.       R     dv  .„. 

consequently  the  equation  (1)  is  the  same  as 


ON    THE    MOMENTS    OF    INERTIA.  187 

W       R2       Wr  dv 

'     ^  g        r    ^    g  Ut 

dv_  R  r  W^—r^  W  ^ 
*'•  lt~   W,  R'^  +  W7^  ^' 
which  expresses  the  accelerative  force  on  the  ascending  weight,  and 
therefore  for  the  velocity  and  space  we  have 

_  RrW,  —  r^  W  _   R  r  W,  —  r^V 

^~    W,  RM-  Wr  ^^'  *  ~  2(W,R^+ Wr)^   * 
For  the  acceleration  of  the  descending  weight  we  have,  in  virtue 

.      ,  ,    dv,       R^  W,  — R  r  W 

of  the  equation  (2),  ^  =    ^y  r.  ^  ^y,..   S 

R2  W,  —  R  r  W  R2  W,  —  Rr  W    ^, 


•  ^  ^  ~    W,  R2  -f  VVr^  *   '     ^       2  (W,  R'^  +  Wr' 
These  two  examples  may  suffice  for  the  present  to  illustrate  the 
application  of  D'Alembert's  principle.     We  shall  have  frequent  oc- 
casion to  refer  to  it  again  in  the  course  of  the  following  chapters. 


CHAPTER  III. 


ON  THE  MOMENTS  OF  INERTIA. 


(155.)  In  treating  of  the  rotation  of  a  solid  body,  we  shall  always 
have  to  take  account  of  that  portion  of  the  impressed  forces  Avhich 
must  necessarily  be  employed  in  overcoming  the  inertia  of  the 
system,  and  which,  therefore,  are  not  effective  in  producing  motion. 
In  a  body  free  to  move  in  any  direction,  the  inertia  to  be  over- 
come by  the  impressed  force  before  motion  can  ensue,  is  obvi- 
ously as  the  mass  to  be  moved ;  but  when  the  body  is  compelled 
to  turn  about  an  axis,  the  amount  of  inertia  will  obviously  vary 
with  its  distance  from  that  axis.  This  resistance  to  motion  about 
any  axis  is  called  the  moment  of  inertia  of  the  system  with  respect 
to  that  axis :  and  we  shall  see  in  the  next  chapter,  that  this  mo- 
ment is  expressed  by  the  sum  of  the  products  of  all  the  particles 
of  the  body  into  the  squares  of  their  respective  distances  from  the 
axis  of  rotation;  that  is,  m,  m',  m",  &c.  denoting  the  component 
particles  of  the  mass  M,  and  r,  r',  r",  &c.  their  respective  dis- 
tances from  the  axis  of  rotation ;  then,  using  the  rotation  already 
employed  at  page  57,  we  shall  prove  that,  with  respect  to  that  axis, 
mom,ent  of  inertia  =  2  (mr^). 

At  present  we  shall  apply  ourselves  merely  to  the  determination 
of  this  expression  in  particular  cases,  as  preparatory  to  the  theory 


188  ELEMENTS    OF    DYNAMICS. 

of  rotation,  to  be  delivered  in  the  succeeding  cliapters.  But  to 
render  the  expression  suitable  for  calculation,  we  must  change  it  into 
the  form  fr^dM,  to  which  it  is  shown  to  be  equivalent  by  precisely 
ihe  same  reasoning  as  that  employed  at  art.  41,  to  which  the  stu- 
dent may  turn. 

If  the  whole  mass  M  of  the  revolving  body  could  be  collected 
into  a  single  point  at  some  distance  k  from  the  axis,  then  the 
moment  of  inertia  of  the  system,  that  is  of  this  point,  would  be 
simply  k^  M,  and  this  will  be  the  same  as  the  moment  which  ac- 
tually has  place,  provided  we  so  determine  k  that 

k^M=fr^dM.:k=^-'—^, 

so  that  if  we  can  determine  A-,  which  is  called  the  radius  of  gyra- 
tion,  we  shall  at  the  same  time  know  the  moment  of  inertia. 

It  is  frequently  of  consequence  to  know  what  mass  M'  ought  to 
be  placed  at  any  proposed  distance  k'  from  the  axis,  so  that  the 
moment  of  inertia  of  the  point  thus  loaded  may  equal  that  of  the 
mass  M.     In  order  to  this,  we  must  evidently  have  A:'-M'^A-'M 

.-.  M'=f-.  M. 

1.  To  determine  the  radius  of  gyration  of  a  slender  rod  of  length 
/  revolving  about  its  extremity. 

Putting  r  for  any  distance  measured  on  the  line  from  the  axis, 

T^  dr  r' 

we  have  by  the  formula  k=y/— — =  v/^j  that  is  for  the  whole 

line,  or  when  r=l,  k=l .^ I  =•57735  I. 

If  the  rod  revolve  about  any  point  in  it  of  which  the  distances  from 
the  extremities  are  a  and  b,  then,  by  taking  the  above  integral  be- 
tween the  limits  r  =  —  b,  r=^a,  we  have,  for  tlie  line  a-\-b, 
_     I   «^  +  63 

2.  To  determine  the  radius  of  gyration  of  a  circle  revolving  ab- 
out an  axis  through  the  centre,  and  perpendicular  to  its  plane. 

Putting  R  for  the  radius  of  the  circle,  its  area  will  be  ft  R%  also 
at  the  distance  r  from  the  axis  the  area  is  n  r^,  therefore,  in  this 

J,,       ^         ,  ,7         Artfr^dr         F"       ^ 

case,  a  M  =  2  rt  rdr,  so  that  «=v — i— - — :=      ^  .  — -, 

rt  K-         \         K^ 

that  is,  when  r=R,  A:=R  y/k^=l  R  ^/'Z,  and  the  result  would  ob- 
viously be  the  same  for  a  right  cylinder  revolving  round  its  axis. 
The  moment  of  inertia  is  A:'^M  =  5  jt  R*. 

3.  To  determine  the  radius  of  gyration  of  a  circular  annulus  or 
flat  ring,  the  axis  of  rotation  passing  through  the  common  centre 
of  the  circles  perpendicular  to  their  plane. 


ON   THE   MOMENTS    OF    INERTIA.  189 

Calling  the  radii  of  the  inner  and  outer  circles  R  and  R',  we 
have,  for  the  area  of  the  annulus,  the  expression  n  R'^  — rt  R^ 

If,  instead  of  the  radius  R',  we  take  any  variable  radius  r,  inter- 
mediate between  R  and  R',  then  the  expression  for  the  annulus  will 
be  M=rtr''— rtR''  .-.  dM=2Krdr 

_  2fr^dr  __^  _r| 

■"      "  ~  R'^_R^  ~2  (R'^-R^y 
and  this,  taken  between  the  proposed  limits  of  r,  that  is  from  r=R 

..=«.,.. .=^-^_«;-....=.jw.(,, 

when  R=0  the  annulus  becomes  a  circle,  and  the  expression  for 
k  agrees  with  that  in  last  example  ;  when  R'=  R  the  expression  is 
k=R  ....  (2),  which  applies  when  the  annulus  becomes  merely 
a  circumference.  It  is  easy  to  see  that  the  expressions  (1)  would 
remain  tlie  same  for  a  cylindric  shell  or  wheel  whose  thickness  is 
R' — R  revolving  about  its  axis,  and  the  expression  (2)  applies  when 
the  shell  is  of  insensible  thickness. 

4.  To  determine  the  radius  of  gyration  of  the  circumference  of  a 
circle  revolving  about  its  diameter. 

The  distance  of  any  point  (x,  y)  from  the  axis  of  rotation  is  y, 
the  origin  being  at  the  centre,  and  the  proposed  diameter  being 
taken  for  the  axis  of  x.    Moreover,  since 

J-——,  we  have  dM  =  ds=  -dx  .'.  y'^  dM.=  Rydx 
ax     y  y 

,,       'Rfydx     R  X  area  of  circle       ?<  R^  ,  -P2 

s  circumference         2 « R 

The  student  cannot  fail  to  have  remarked  the  similarity  between 
these  problems,  and  those  for  finding  the  centre  of  gravity,  and  he 
must  further  observe  that  in  determining  the  distance  k  of  the  centre 
of  gyration  from  an  axis,  the  same  regard  must  be  paid  to  the  man- 
ner in  which  d  M  is  taken  that  was  necessary  in  determining  the 
centre  of  gravity,  as  the  form  of  this  diflerential  will  vary  with  the 
position  of  the  axis  from  which  k  is  measured.  If  the  body  is  a 
plane  area,  and  if,  moreover,  the  axis  from  which  k  is  measured  is 
in  that  plane,  then,  taking  this  axis  for  that  of  x,  and  putting  (/M 
under  the  general  form  dM=fdxdy,  as  at  page  64,  the  expression 
for  A;^M  will  be,  since  what  we  have  before  called  r  is  now  y, 

km=ffy^dxdy=f^. 

Or  this  expression  for  the  moment  of  inertia  of  a  plane  area,  with 
respect  to  an  axis  in  it,  may  be  deduced  from  considering  the  area 
to  be  generated  by  the  ordinate  y  moving  along  the  axis  of  x ;  for 
as  y  generates  the  whole  area,  so  must  the  momentum  of  y  generate 


190  ELEMENTS  OF   DYNAMICS. 

the  whole  momentum ;  in  the  one  case  the  generatrix  is  y,  and 
the  quantity  generated  fydx;  in  the  other  case  the  generatrix 
(see  ex.  1,) 

is  ^  and  therefore  the  quantity  generated  must  be  /  ~ — . 
3  *J      o 

5.  Applying  this  formula,  to  the  circle  revolving  about  its  dia- 
meter or  axis  of  x,  we  have,  from  the  equation  of  the  circle, 
y^^2rx — x^  .'.  2ydy={2r — 2x)  dx 

.-.  k=M  =  -r—r-'^^j!M==^rtr*(Int.  Calc,  p.  42), 

this  is  for  the  semicircle  ;  but  if  instead  of  integrating  from  y= — r 
to  y^r,  we  had  integrated  from  y= — 0  to  y^O,  we  should  have 
had  for  the  whole  circle  A:'^M  =  4  nr*;  as  in  the  whole  circle  M 
is  twice  as  great  as  in  the  semicircle,  k^  is  the  same  for  both,  viz. 

To  determine  the  moment  of  inertia  of  a  solid  of  revolution, 
with  respect  to  the  fixed  axis,  it  will  be  most  convenient  to  view 
the  solid  as  generated  by  the  motion  of  a  circle  which  continues  al- 
ways perpendicular  to  the  fixed  axis  while  the  centre  describes  this 
axis,  as  explained  at  page  144  of  the  Integral  Calculus.  In  this 
point  of  view,  we  may  obviously  consider  the  whole  moment  of  in- 
ertia to  be  generated  by  the  moment  of  inertia  of  the  generating 
circle ;  calling  the  variable  radius  of  tliis  y,  the  fixed  axis  being 
that  of  X,  the  generating  moment  will,  by  ex.  2,  he  i  ny*;  hence 
the  whole  moment  generated  must  be  k  2M  =  5  ttfy*dx. 

6.  Suppose  tlie  solid  were  a  sphere  of  radius  a,  then 
2/2  -=  2  (IX  —  X-  .•.  y*  dx  =  (2  ax  —  x^)^  dx 

.•.A:2  M  =  hnf{2ax—x^)    ''dx=H  x^  {^  a^  —  h  ax  -^^\x^), 
and  this  integral  between  the  limits  x  =  0,  a?  =  2a,  gives  for  the 

8  2 

whole  sphere  k^  'M  =  -—  rt  a' ,  .• .  k'^  =—  a^ .    In  like  manner  we 
^  15  5 

should  find  for  a  cone  revolving  about  its  axis  A;2=  ~^r^,  r  being 

the  radius  of  the  base. 

For  a  paraboloid  the  expression  is  k^  =  l  r^. 

(156.)  It  is  useful  to  know  how  to  find  the  moment  of  inertia, 
with  respect  to  an  axis,  by  means  of  the  known  moment  with 
respect  to  some  other  axis  parallel  to  it.  We  shall,  therefore,  now 
show  how  this  is  to  be  done. 

Let  AZ  be  the  axis  (fig.  112,)  for  which  the  moment  of  inertia 
isy>*(ZM,  and  let  A'Z'  be  the  axis  parallel  to  it  for  which  the  mo- 
ment of  inertia  fr'^dM.  of  the  same  mass  M  is  to  be  determined.  As- 

X  il-^.    .'.  «  :  r: ' .  'v ''f  /  s:  ■'■  i ' '-  '^^^ 


ON    THE    MOMENTS    OF    INERTIA.  191 

sume  AZ  to  be  the  axis  ofz,  and  AX,  AY  to  be  those  of  a:  andy;  then 
for  every  particle  m  of  the  body,  the  corresponding  value  of  Z?n  or 
of  its  projection  Am'  is  r^=x''-\-y^.  In  like  manner,  the  distance 
of  the  two  axes  being  a,  if  we  call  the  co-ordinates  AA',  A'B  of  this 
axis  a,  i3,  we  shall  have  a^  =  a^  +  Z^'^-  Now  the  distance  of  the 
particle  m  from  A'  Z',  that  is  of  the  point  (x,  y)  from  the  point 
(a,  /3),  is 

r'^={x—ay+{y-~^y 

—  x^-\-y"-—1  a  X—2  ^y-\-a^-\-^'^ 
=  r^ — 2  a  X — 2  ^y-\-a^,  therefore 
/r'2 rfM=/r2 dM—2  ajx  rfM— 2 ^  fy  dM-^-a^M; 
or,  putting  X  and  Y  for  the  co-ordinates  of  the  centre  a  gravity  of 
the  system,  Ave  have  (art.  41,) 

fr'^ dM=fr^ (M+a^M  —  2M  (aX-f^Y). 
When  the  original  axis  passes  through  the  centre  of  gravity,  then, 
since  Y=0  and  X^O,  the  formula  becomes 

fr'"  dM=fr"-dM-\-a''M,  or  M  A:'=^=M  [k^+d") ; 
hence  to  the  moment  of  inertia,  estimated  from  an  axis  through  the 
centre  of  gravity,  we   must  add  the  product  of  the  mass  by  the 
square  of  its  distance  from  the  new  axis,  and  the  sum  will  be  the 
new  moment  of  inertia. 

The  foregoing  expression  for  the  moment  of  inertia  about  any 
axis,  shows  that  of  all  axes  taken  parallel  to  each  other,  that  which 
passes  through  the  centre  of  gravity  of  the  body  will  be  the  one  in 
reference  to  which  the  moment  of  inertia  is  the  least ;  k  is  in  this 
case  called  the  principal  radius  of  gyration. 

1.  Take  a  straight  line  or  slender  rod  I  revolving  about  an  axis 
at  the  distance  a  from  the  middle  ;  then,  by  the  first  example  of  last 
article,  the  moment  about  the  middle  is  •y'j  P,  therefore  the  moment 
about  the  proposed  axis  is  k'^  M=y^2  l^-{-a^  I. 

2.  Let  a  circle  revolve  about  any  line  in  its  plane. 

The  moment  round  a  diameter  parallel  to  the  line  is,  by  ex.  5, 
irtr*,and  consequently,  calling  the  distance  of  this  diameter  from 
the  proposed  line  a,  we  have 

k^  M=k  rt  r*+a^  M  .-.  A;2==|  j-^^a^ 
The  result  is  necessarily  the  same  when  the  proposed  axis  is  out 
of  the  plane  of  the  circle. 

3.  Required  the  moment  of  inertia  of  a  cone  revolving  about  an 
axis  through  the  vertex,  and  perpendicular  to  the  axis  of  the  cone 
(fig.  113). 

Let  VA  be  to  AB  as  1  to  n,  then  calling  the  distance  VA'  of  any 
variable  section  x,  we  have  A'  P=/ia;  for  the  radius  of  that  section, 
and  for  its  moment  of  inertia  the  expression,  as  found  in  last  ex- 
ample, is 


192  ELEMENTS  OF  DYNAMICS. 

i  rt7i*  x*-\-x'^M=i  rtn*x*+rtn^x*=7tn^{4  n«+l)a;*; 
hence,  for  a  solid  generated  by  this  circle,  the  moment  is 

which  applies  to  the  cone  of  altitude  x. 


CHAPTER  IV. 

ON    THE    ROTATION    OF    A    SOLID    BODY    ABOUT    A    FIXED    AXIS. 

(157.)  Suppose  a  body  of  a  known  form  and  mass  to  be  freely 
moveable  about  a  fixed  axis,  passing  through  A  (fig.  114),  and  per- 
pendicular to  the  plane  on  which  the  figure  is  represented. 

Suppose  an  impulse  or  shock  to  be  given  to  the  body  thus  cir- 
cumstanced, and  that  it  is  in  the  direction  BC,  perpendicular  to  the 
plane  AB,  drawn  through  the  fixed  axis ;  if  the  impulsion  were 
oblique  to  this  plane  we  should  only  have  to  take  account  of  that 
component  of  it  which  is  perpendicular  to  the  plane,  because  the 
other  being  directed  towards  the  fixed  axis,  would  be  counteracted 
by  its  resistance,  and  its  effect  destroyed. 

If,  when  the  impulse  was  given  at  B,  we  conceive  the  mass  to 
have  been  perfectly  free,  and  to  have  .been  concentrated  in  the  point 
B,  or,  which  is  the  same  thing,  if  we  conceive  the  impulse  to  have 
been  directed  towards  the  centre  of  gravity  of  an  equal  mass  M, 
perfectly  free,  then,  v  being  the  velocity  communicated,  Mv  would 
be  the  momentum  communicated;  this  expression,  therefore,  pro- 
perly represents  the  intensity  of  the  impulse  or  the  force  impressed 
on  the  systems  (150).  The  effect  of  this  impulse  upon  the  parti- 
cles of  the  body,  under  consideration,  is  to  cause  each  to  describe  a 
similar  arc  of  a  circle  in  the  same  time,  the  radius  of  any  one.  being 
Am=r  ;  each  particle  will,  therefore,  move  with  the  same  angular 
velocity  about  the  fixed  axis,  and  this,  therefore,  may  be  called  the 
angular  velocity  of  the  whole  mass ;  let  it  be  represented  by  w. 
Now  the  only  force  impressed  on  the  system  is  Mi»,  acting  at  B, 
and  the  forces  witli  which  the  component  particles  vi,  in',  in".  Sic. 
move,  arise  from  their  mutual  connexion  with  each  other ;  we  may, 
therefore,  apply  to  this  inquiry  the  principle  of  D'Alembert,  and 
thus  obtain  an  equation  between  the  impressed  and  the  actual 
forces.  As  the  particles  all  have  the  common  angular  velocity  u, 
their  actual  velocities  are  rw,  r'w,  r"  u.  Sic.  and,  consequently,  their 
moving  forces,  or  momenta,  mru,,  m'  r'  u,  m"  r"  w,  &;c.,  and  these 
are  the  forces  which  acting  at  the  distances  r,  r',  r",  &c.  from  the 
axis  of  rotation,  must  equilibrate  with  the  force  Mv,  acting  at  the 
distance  R=AB,  when  this  latter  acts  in  the  opposite  direction ; 


CENTRE    OF    OSCILLATION.  193 

consequently,  multiplying  each  force  by  its  distance  from  the  axis, 
we  must  have  the  equation 

lVTT?i) 

mr^i^+m'  r'  ^  o>+m"  r"  ^  ui-\-ikc.=MRv  .-.  co=— — ; — ■.  (1) ; 

2  {r-  m)    ^  ' 

which  implies  that  the  angular  velocity  is  equal  to  the  moment  of 
the  applied  force,  divided  by  the  moment  of  inertia,  just  as  the 
linear  velocity  v,  which  the  same  force  is  competent  to  produce  on 
an  equal  mass,  is  expressed  by  that  force,  M.v,  divided  by  the  mass 
moved,  by  which  mass  the  inertia  of  the  body  is  always  repre- 
sented :  so  that  as  M  denotes  the  resistance  to  progressive  motion, 
X  (r^in)  denotes  the  resistance  to  angular  motion.  This  expression, 
therefore,  is  fitly  called  the  moment  of  inertia  of  the  revolving  body. 
(158.)  Let  us  now  suppose  that  instead  of  an  impulse  giving  ro- 
tation to  the  body,  every  particle  of  it  is  actuated  by  an  accelerative 
force ;  these  forces  may  be  all  different,  but,  as  before,  we  shall 
consider  them  as  acting  in  planes  perpendicular  to  the  fixed  axis. 
Calling  the  several  forces  acting  at  m,  m',  m",  &c.  F,  F',  F",  &c. 
and /J,  p',p",  the  perpendiculars  from  the  axis  on  their  directions; 
the   applied  motive  forces  will  be  mF,  m' F',  m"  F",  &c.  and  u 

being  the  angular  velocity,  or  rw  the  absolute  velocity  of  m,  r  — 

dcj 
will  be  its  acceleration,  so  that  the  actual  motive  forces  are  mr  —  , 

m'  r'  —,  m"  r"  — ,  &c. ;  hence,  by  D'Alembert's  principle,  these 

forces,  taken  in  the  reverse  direction,  balance  the  former,  so  that 
their  moments  give  the  equation 

(«ir^+m'  r'^+m"  r'^)—=mFp-\-m'  F'p'-{-m"  F" p"-\-&,c. 

.  (/^^S  (Fpm)  _ 

"dt       2  {r"-  m)  ^^  ' 

that  is,  as  before,  the  angular  acceleration  is  equal  to  the  moment 
of  the  applied  moving  forces,  divided  by  the  moment  of  inertia, 
just  as  the  linear  acceleration  which  the  same  forces  sF  •  2w  is 
competent  to  produce  in  an  equal  mass,  2?72  is  expressed  by  that 
force  divided  by  the  mass  moved,  that  is,  by  2»i. 

Let  us  now  apply  the  result  just  obtained  to  the  theory  of  the 
compound  pendulum. 

Centre  of  Oscillation  of  a  vibrating  Body. 

(159.)  When  a  heavy  body  vibrates  about  a  horizontal  axis,  by 
the  force  of  gravity,  the  mass  is  considered  as  a  compound  pendu- 
R  25 


194  ELEMENTS  OF  DYNAMICS. 

lum,  and  it  is  an  important  problem  to  determine  what  must  be  the 
length  of  a  simple  pendulum  which  shall  perform  its  oscillations 
in  the  same  time  about  the  same  axis  of  suspension  ;  or  rather 
to  determine  what  simple  pendulum  must  be  substi luted  for  the 
compound  mass,  in  order  that  the  vibrations  of  both  may  be  the 
same  as  regards  velocity,  acceleration,  and  time.  'J'he  foregoing 
general  formula  enables  us  to  solve  this  problem  ;  for  as  the  acce- 
lerative  force  acting  upon  every  particle  of  the  mass  is  the  same, 
viz.  F^g,  the   formula,  as  applied  to   this  case,  may  be  written 

Now  let  G  (fig.  115,)  be  the  centre  of  gravity  of  the  oscillating 
body,  M,  and  AB  a  vertical  plane  through  the  horizontal  axis  of 
suspension,  which  axis  we  here  suppose  to  l>e  perpendicular  to  the 
plane  of  the  paper ;  let  GP  be  perpendicular  to  this  plane  when 
the  body  is  in  any  proposed  position,  that  is,  when  the  plane  AM, 
through  the  centre  of  gravity  and  perpendicular  to  the  plane  of  the 
paper,  makes  any  angle  GAB=0  with  the  plane  AB ;  then  we 
know,  by  the  theory  of  the  centre  of  gravity  (41),  that 

MxGV=mp-\-m'p' +m"p"-\-,  &c. 
that  is,  putting  a  for  the  distance  AG  of  the  axis  from  the  centre  of 
gravity,  since  GP  =  a  sin.  e  ;    M  .  a  sin.  0  =  S  {mp),  substituting, 
therefore,  this  value  in  the  numerator  of  the  expression  (1)  above, 
we  have 

rfu     M  .  a  sin.  6     M  .  «  sin.  B     a  sin.  6  . 

l/?^"T(r2wy~^M|F+a2^==pq:^  ••••()' 

where  k  represents  the  radius  of  gyration,  in  reference  to  an  axis 
parallel  to  that  of  suspension,  and  passing  through  the  centre  of 
gravity  of  the  body,  and  where  a  is  the  distance  of  these  two  axes. 
Suppose  now  that  the  mass  were  removed,  and  that  in  its  stead 
there  were  suspended  a  single  particle  at  the  distance  /  from  the 
axis,  then  2  (r^m)  would  become  simply  I'^m,  and  in  order  that  the 
angular  acceleration  of  this  simple  pendulum  may  be  the  same  as 
that  of  the  mass,  for  which  it  has  been  substituted,  we  must  deter- 
mine /  from  the  equation, 

M  .  a  sin.  e     ml  sin.  6    -         a  1 

"^  {r^m)    ~     Wm      '  '  2  (r^n)  ~~  7 

this,  therefore,  expresses  the  length  of  the  simple  pendulum,  whose 
vibrations  will  agree  with  that  of  the  compound  pendulum :  it  is 
equal  to  the  square  of  the  radius  of  gyration  measured  from  the  axis, 
divided  by  the  distance  between  the  axis  and  centre  of  gravity. 


CENTRE    OF    OSCILLATION.  195 

That  point  in  the  body,  wliich  is  at  tlie  distance  /  from  the  axis,  is 

called  the  centre  of  osdllation  of  the  compound  pendulum.    There 

are,  it  is  evident,  an  infinite  number  of  such  centres,  or  of  points  at 

the  distance  /  from  the  axis,  but  we  here  more  especially  mean  that 

point  which  is  in  the  line  passing  through  the  point  of  suspension 

and  the  centre  of  gravity  of  the  mass. 

As  OG  is  the  distance  of  the  centre  of  gravity  G,  from  the  centre 

of  oscillation  O,  it  follows,  from  the  equation  (3),  that  the  distance 

A- 2 
between  these  two  centres  is  0G= —  ....  (4);  therefore,  since 

a 

so  long  as  the  plane  of  the  body's  vibration  remains  the  same  h  must 

remain  the  same,  it  follows  that  with  this  condition  the  distances 

between  the  centres  of  gravity  and  oscillation  are  inversely  as  the 

distanrps  between  the  centre  of  gravity  and  point  of  suspension. 

Moreover,  if  the  centre  of  oscillation  0  were  made  the  point  of 
suspension,  then  to  find  the  corresponding  value  of  /  we  must  put 
ttfe  value  (4)  for  a  in  (3)  which  substitution,  as  it  alters  not  the 
value  of  /,  shows  us  that,  provided  we  do  not  alter  the  plane  of  the 
body's  vibration,  the  centre  of  oscillation  and  point  of  suspension 
are  convertible  ;  that  is,  if  we  convert  the  centre  of  oscillation  into 
the  point  of  suspension,  the  point  of  suspension  will  become  the 
centre  of  oscillation. 

Also  the  distances  of  the  axis  of  suspension  from  the  centres  of 
gravity,  of  gyration,  and  of  oscillation,  are  in  continued  proportion, 

for  a  :  \/\a^-\-k^\  : :  ^\a^-\-h^\  :  a-\ — .     We  may,  likewise,  infer 

from  the  value  of  Z,  what  the  distance  a  of  the  centre  of  gravity  from 

the  axis  must  be,  in  order  that  for  the  same   plane  of  vibration  the 

time  in  which  the  body  performs  its  oscillations  may  be  the  least 

possible  ;  for  as  the  equivalent  simple  pendulum  will  vibrate  the 

quicker  the  shorter  it  is,  we  shall  merely  have  to  determine  a  so 

that  we  may  have 

A:^  .   .  dl      ^       ¥ 

l=a-i — =a  mimmurn.     .-.  -7-  =  !' =0  .-.  a=k: 

a  da  a'^ 

so  that  the  axis  of  suspension  must  pass  through  the  principal  cen- 
tre of  gyration.  Several  other  particulars  might  be  deduced  from 
the  foregoing  investigation,  but  we  shall  mention  here  but  one  more, 
which  is  that  if  the  compound  mass  consist  of  several  distinct  bodies, 
the  centre  of  oscillation  of  the  whole  will  be  found  by  taking  the 
continued  product  of  each  mass  into  the  respective  distances  of  its 
centres  of  oscillation  and  of  gravity,  from  the  axis  of  suspension, 
adding  the  products  together,  and  dividing  the  sum  by  the  product 
of  the  whole  system  into  the  distance  of  the  common  centre  of 
gravity  from  the  axis. 


196  ELEMENTS    OF    DYNAMICS. 

For  calling  the  several  masses  M,  M',  &c.  and  the  length  of  the 
equivalent  pendulum  L,  we  have,  by  the  preceding  theory, 


1  = 


M  .  a 

M'  {a'^-j-k'") 


^~      M'.a' 


^.•.2(/.M.a)=slM(o'+A-^)|. 


&c.         &c. 

As  the  second  member  of  this  equation  expresses  the  moment  of 
inertia  of  the  whole  system,  it  follows  that 

^-  2(M.a)  ••••^^^ 
which  establishes  the  proposition,  since  the  denominator  of  this 
fraction  is  equal  to  the  whole  compound  mass  into  the  distance  of 
its  centre  of  gravity  from  the  axis.  This  same  formula  will,  obvi- 
ously, serve  for  any  vibrating  mass  when  we  find  it  convenient  to 
consider  it  in  separate  parts. 

(160.)  We  shall  now  give  a  few  examples  of  the  determination 
of  the  centre  of  oscillation  in  different  bodies. 

1.  To  determine  the  centre  of  oscillation  of  a  slender  rod  or 
straight  line  suspended  at  any  point. 

Let  a,  b,  be  the  lengths,  on  contrary  sides,  of  the  point  .of  suspen- 

sion,  then  (155)  for  k^  we  have,  k^=— — — — -. 
^       '  a-\-h 

Again,  the  centre  of  gravity  being  in  the  middle  of  the  line,  its 
distance  from  the  point  of  suspension  is  ^  (o  —  h)\  hence  (3), 
,_ 3  (o'  +  ^")  _2jo=— _o^+^) 
—  A  (a3_6i)—      3(«  — 6)      ■ 
If  the  rod  is  suspended  at  its  extremity,  then  6=0,  and  l=\a.,  or 
two  thirds  the  length.     If  it  is  suspended  at  its  middle,  then  a-=b, 
and  /=  Gc,  that  is,  the  centre  of  oscillation  is  at  an  infinite  distance, 
and,  therefore,  to  perform  one  vibration  would  require  an  infinite 
length  of  time,  and  this  is  tlie  same  as  saying  that  no  vibration  at 
all  could  take  place  ;  indeed,  in  whatever  position  about  the  centre 
of  motion  the  rod  be  placed,  it  would  obviously  rest  there,  seeing 
tliat  its  centre  of  gravity  would  be  supported. 

If  6  =  do,  then  l:=a,  or  two  thirds  the  whole  length,  the  same 
distance  as  when  the  rod  is  suspended  at  its  extremity  ;  so  that  in 
both  these  cases  the  oscillations  will  be  performed  in  the  same  time, 
as,  indeed,  ought  to  be  the  case,  because  the  centres  of  suspension 
and  of  oscillation  are  merely  interchanged,  (p.  194.) 

2.  To  determine  the  centre  of  oscillation  of  an  angular  pendulum 
(fig.  116,)  composed  of  two  equal  slender  rods  AB,  AC. 

Bisect  the  arms  AB,  AC,  in  ^  and  y ;  these  points  will  be  their 


CENTRE    OF    OSCILLATION.  197 

centres  of  gravity ;  bisect  the  line  joining  them  in  G,  and  this  will 
be  the  centre  of  gravity  of  the  system,  and  AG^Ag*  sin.  g,  or  AG 
=  5acos.  e. 

Again,  the  distance  of  A  from  the  centre  of  oscillation  of  the  part 
AB  or  of  AC,  is  equal  to  |  AB  =  |fl,  therefore,  by  the  formula  (5), 

L=    ^'^    , ^—=%  a  sec.  6 ; 

2  a  .  i  a  cos.  9 

hence  because  L,  the  length  of  the  equivalent  simple  pendulum,  in- 
creases with  6,  it  follows  that  the  time  of  vibration  of  an  angular 
pendulum  may  be  increased  without  limit,  by  merely  increasing  the 
angle  between  the  arms  ;  when  6=0,  that  is,  when  the  arms  close, 
and  form  but  one  rod  :=  a,  the  corresponding  value  of  L  is  |a,  and 
when  0=90°  or  when  the  arms  open  into  a  straight  line,  L  is  in- 
finite. 

When  the  time  of  vibration  is  given,  we  may  easily  determine 
the  corresponding  angle  of  the  arms  when  their  lengths  are  known, 
for,  from  the  given  time,  L  will  be  determined,  and  thence 

3L 

sec.  e=- — . 
2  a 

Thus  if  the  time  is  one  second,  then  L  =395,  and  if  «=15  inches 

sec.  0=75°  •  U'i  .-.  /_  BAC  =  150°  •  23'. 

3.  To 'determine  the  centre  of  oscillation  of  a  sphere. 

Let  r  be  the  radius  of  the  sphere,  then  the  mass  is  |  rtr^,  and  the 
square  of  the  principal  radius  of  gpation  is  (p.  189-190,)  A:^=|r^ ;  hence 

(  =  «+-=»+-. -....(I); 

a  being  the  distance  of  the  centre  of  the  sphere  from  the  point  of 
suspension. 

If  the  axis  of  suspension  were  a  tangent  to  the  sphere,  we  sliould 
have  r  =  a,  therefore,  in  that  case,  I  =  r  -\-  fr. 

From  the  expression  (1)  we  get  for  a,  w^hen  Z  is  given, 

a  =  n  ±  x/  [W—ir^; 
so  that  a  sphere  may  be  suspended  at  two  different  distances  from 
the  centre,  and  yet  vibrate  in  the  same  proposed  time  ;  if,  however, 
I  /*  =  I  7-^,  that  is,  if  the  time  of  vibration  is  to  be  that  which  be- 
longs to  the  simple  pendulum,  Z  =2?- ^|,  then  there  is  but  one 
suitable  distance  for  the  centre  from  the  axis  of  suspension,  viz. 
the  distance  a  =  r  ^  |. 

4.  Suppose  the  bob  of  a  clock  pendulum  to  consist  of  two 
spheric  segments  joined  at  their  bases  :  to  determine  the  distance 
of  the  centre  of  oscillation  from  the  centre  of  the  bob. 

Let  r  be  the  radius  of  the  sphere,  and  x  the  height  of  one  of  the 
segments,  then  the  moment  of  inertia  of  the  two  segments  is  (ex.  6, 
r2 


198  ELEMENTS  OF  DYNAMICS. 

p.  190),  MA-«  =  2  ;t  x^  (I  r«  — 5  rx  +j\x''),  and  the  volume  of  the 
two  segments  is 

M  =  2  ,  ..  (r-  J  .)  .-.  i'=  ^^S^^ 
^  '       a  a  (r — j  x) 

and  this  is   the  distance  of  the  centre  of  gravity  of  the  bob  from 

the  centre  of  oscillation. 

If  instead  of  the  radius  r  of  the  sphere,  the  radius  r'  of  the  base 

of  each  segment  is  given ;  then  since  r'"  =  (2  r — x)  x 

.-.  r=  — ■ and,  by  substitution,  -  = 7..  ,„  , — rr , 

a  added  to  either  of  these  expressions  will  give  the  distance  of  the 
centre  of  oscillation  from  the  point  of  suspension. 

5.  To  determine  the  centre  of  oscillation  of  a  cone  suspended 
at  its  vertex. 

Putting  (as  in  ex.  3,  page  191),  x  for  the  altitude  of  the  cone, 
and  nx  for  the  radius  of  its  base,  we  have,  for  the  moment  of  inertia, 
the  expression  -}  rt  7i^  x^  (?  n^  -^  1),  also  the  distance  of  the  centre 
of  gravity  from  the  vertex  is  (4G,)  I  x,  consequently, 

^^W^x^  {kn-  -\-\)  ^\  ^n-  x^  {k  n-  +  \)^  ,  ^^,^      ,^ 
M.ix  h  x.nn^x^'.ix         *  *    ' 

6.  In  a  similar  manner  is  found  for  the  centre  of  oscillation  of  a 
circle  vibrating  edgewise,  and  suspended  at  the  distance  <t  from  its 

centre,  /  =  a  +  :;-• 
2« 

7.  And  when  the  circle  vibrates  flatwise  I  =  a  -\-  ■—. 

4a 

On  the  Centre  of  Percussion. 

(161.)  When  a  solid  body  revolves  about  a  fixed  axis,  the  {ore4 
with  which  it  would  strike  a  fixed  obstacle  would  vary  with  the 
.situation  of  the  point  of  impact,  as  also  would  the  shock  received 
by  the  immoveable  axis  ;  but  there  is  a  point  at  which,  if  the  im- 
pact take  place,  the  obstacle  will  receive  the  whole  force  of  the 
moving  body,  and  the  axis  will  receive  none,  so  that,  if  at  the  in- 
stant of  impact  the  axis  were  to  be  annihilated,  the  body  would 
still  remain  at  rest.  This  point  is  called  the  centre  of  percussion. 
In  order  to  ascertain  the  situation  of  this  centre,  let  us  refer  to  the 
expression  (1)  for  <o,  u  being  here  considered  to  represent  the  an- 
gular velocity  at  the  instant  of  impact ;  let  also  the  moving  force 
due  to  the  impact,  and  which  we  before  represented  by  Mv,  be 
called/,  and  the  distance  of  the  direction  of  impact  from  the  axis 

<d  2  (r^  in) 
D,  the  formula  referred  to  then  gives  /  =  ^^ -;  this  ex- 


CENTRE  OF  SPONTANEOUS  ROTATION.  199 

presses,  therefore,  the  force  of  percussion  at  the  distance  D.  Now 
if  this  force  is  equal  to  that  of  the  whole  moving  mass,  no  portion 
of  it  will  be  expended  in  straining  the  axis,  and  D  will  in  that  case 
measure  the  distance  of  the  centre  of  percussion  from  the  axis. 

To  determine  what  the  whole  force  of  the  revolving  body  is,  Ave 
must  add  together  the  forces  due  to  the  component  particles  ;  that 
due  to  any  one,  m,  is  mru> ;  hence,  calling  the  whole  force  F,  we 
have 

^     ^  ^     ^  D  2  rm) 

consequently  the  distance  of  the  centre  of  percussion  from  the  axis 

is  equal  to  the  distance  of  the  centre  of  oscillation  from  the  axis  ;  if, 
therefore,  the  impact  be  perpendicularly  directed  to  the  plane  pass- 
ing through  the  axis  of  suspension  and  centre  of  gravity,  then  the 
centres  of  percussion  and  oscillation  will  be  in  the  straight  line  par- 
allel to  the  axis  of  suspension. 

If  the  plane  through  the  centre  of  gravity,  and  perpendicular  to 
the  fixed  axis,  divide  the  body  symmetrically,  it  is  plain  that  what- 
ever force  directed  in  this  plane  strike  the  body,  the  axis  may  sutler 
a  direct  shock,  but  it  will  not  be  twisted  ;  in  such  bodies,  therefore, 
the  centre  of  percussion  coincides  with  the  centre  of  oscillation,  be- 
cause at  this  point  the  impact  will  neither  strain  nor  twist  the  axis. 
But  in  other  cases,  impact  at  the  centre  of  oscillation,  although  it 
would  occasion  no  direct  strain  on  the  axis,  may  yet  tend  to  twist 
it,  as  it  is  easy  to  conceive,  (see  Br.  Gregory^ s  Mechanics,  vol.  l.p. 
300  ;  and  Francoeur^s  Mecanique,  p.  352.) 

Between  the  centre  of  gyration  of  a  body  revolving  about  a  fixed 
axis  and  the  centre  of  percussion,  we  may  remark  this  difierence, 
viz.  the  centre  of  gyration  is  the  point  in  which,  if  the  whole  revolv- 
ing mass  were  concentrated,  the  same  angular  motion  would  be 
generated  by  any  force  as  before,  and  therefore,  an  obstacle  meeting 
the  line  which  connects  this  point  with  the  axis  of  motion,  will  be 
struck  with  the  same  force  as  it  would  be  by  the  revolving  mass,  at 
whatever  point  of  the  line  the  impact  take  place.  But  the  centre  of 
percussion  is  that  particular  point  which  would  strike  an  obstacle 
with  the  whole  force  of  the  revolving  mass. 

Of  the  Centre  of  Spontaneous  Rotation. 

(162.)  Intimately  connected  with  the  centre  of  percussion  is  that 
of  spontaneous  rotation. 

We  have  just  seen  that  when  a  body  revolves  on  an  axis,  there 
exists  a  point  at  which  a  fixed  obstacle  would  receive  the  whole  of 
its  force,  so  that  if  this  same  force  were  to  strike  the  body  when  at 


200  ELEMENTS    OF    DYNAMICS. 

rest  in  the  same  point,  it  would  produce  in  it  a  rotatory  motion  round 
the  same  axis,  even  though  that  axis  had  l)ocn  removed ;  the  axis 
about  which  a  quiescent  body,  when  struck  in  a  direction  not  pass- 
ing tlirough  tlie  centre  of  gravity,  thus  spontaneously  revolves,  is 
called  the  axis  of  spontaneous  rotation. 

Instead  then  of  considering  the  body  which  receives  the  impulsion 
to  be  retained  by  a  fixed  axis,  let  us  suppose  it  to  be  perfectly  free ; 
or,  to  render  the  inquiry  the  more  general,  let  us  first  consider  a 
system  of  material  particles  7n,  m' ,  m'\  Sic.  entirely  free  and  uncon- 
nected, and  moving  with  the  parallel  velocities  i-,  v',  v",  &c.  and  let 
us  inquire  what  will  be  the  motion  of  the  centre  of  gravity  of  the 
system. 

Through  this  centre  conceive  a  plane  parallel  to  the  directions  of 
the  impvdsions ;  as,  at  the  commencement  of  the  motion,  the  sum  of 
the  moments  of /n,  ?n',  m",  &c.  with  respect  to  this  plane,  is  0,  (p. 
59,)  and  as  the  bodies  preserve  their  respective  distances  from  the 
plane  througliout  the  motion,  it  follows  that  the  sum  of  the  moments, 
with  respect  to  this  plane,  must  be  the  same  at  every  instant ;  hence, 
the  sum  being  always  0,  the  centre  of  gravity  of  the  system  must  ne- 
cessarily move  always  in  this  plane.  Conceive  another  plane  also  pa- 
rallel to  the  directions  of  the  impulsions,  and  passing,  in  like  manner, 
through  the  centre  of  gravity  of  the  system,  but  making  an  angle  with 
the  former  plane,  then,  as  before  shown,  the  centre  of  gravity  will  al- 
ways move  in  this  plane,  it  must,  therefore,  describe  the  line  of 
intersection  of  these  planes,  that  is,  the  centre  of  gravity  of  the  sys- 
tem describes  a  straight  line  parallel  to  the  directions  of  the  impress- 
ed velocities. 

Conceive  now  a  plane  perpendicular  to  the  directions  of  the  ve- 
locities, and  design  by  e,  e',  e"  Sic.  the  distances  of  the  points  m,  m' 
m",  Sic.  from  this  plane  at  the  commencement  of  motion:  at  the 
end  of  the  time  t",  their  distances  will  be  e-\-vt,  e'-j-uV,  e"-\-v"t. 
Sic. ;  let  us  take  the  moments  with  respect  to  this  plane,  then  a,  r 
being  the  respective  distances  of  the  centre  of  gravity  from  the  plane 
at  the  commencement  of  motion,  and  at  the  end  of  the  time  t",  we 
shall  have  the  following  equations  (41) 

(m-\-m'-\-7n"-\-Sic.)  a=me-\-m'c'-{-7n"e"-\-Sic. 
(yn-f-m' -f  m"-|-&;c.)  .r=m(e-|- i'/)-f  m'(e' -f  u'f) -f&c. 
Subtracting  the  first  from  the  second  we  have 

(77i-fm'+?n"-f-&;c.)  {x — a)={mv-\-m'v'-\-m"v"-{-Sic.)  t, 
which  shows  that  the  space  rr— o,  described  by  the  centre  of  gravity, 
is  proportional  to  the  time  ;  hence  the  centre  of  gravity  of  the  sys- 
tem moves  uniformly.  It  must  be  remembered,  that  in  the  foregoing 
equations  those  values  of  e,  e',  Sic,  of  w,  v',  Sic,  are  taken  nega- 
tively which  are  measured  in  opposite  directions  to  those  considered 
as  positive. 


CENTRE  OF  SPONTANEOUS  ROTATION.  201 

X  — fl 

As is  the  velocity  of  the  centre  of  gravity,  it  follows,  that 

if  the  whole  mass  of  the  system  were  concentrated  there,  its  mo- 
mentum or  quantity  of  motion  would  be 

X  "^^  a  ' 

{m-\-m' -\-m" -\-&Lc.)  =mv^m'v' -\-m"v" -]-&lc. 

by  the  equation  just  deduced  ;  hence  the  second  member  of  this 
equation  represents  the  intensity  of  the  impulsion  that  must  be  ap- 
plied to  the  whole  mass  of  the  system,  when  concentrated  in  the 
centre  of  gravity,  to  make  that  centre  move,  as  it  actually  does ;  we 
conclude,  therefore,  that  the  centre  of  gravity  moves  with  the  same 
velocity  as  if  all  the  impulsions  ivere  immediately  impressed  on  it, 
or,  which  is  the  same  thing,  the  centre  of  gravity  moves  as  if  the 
whole  mass  of  the  system  loere  concentrated  in  it,  and  all  the 
forces  were  applied  to  it  in  directions  parallel  to  those  they  really 
take. 

If  the  impressed  velocities  were  not  parallel,  the  same  thing  would 
also  have  place.  For  if  we  decompose  each  of  them  into  three 
others  parallel  to  rectangular  axes,  we  may  apply  to  each  of  the 
three  groups  of  parallel  velocities  the  foregoing  reasoning,  and  thence 
infer  that  the  centre  of  gravity  would  move  in  the  direction  of  each 
axis,  as  if  the  forces  parallel  to  that  axis  were  immediately  applied 
to  that  centre  ;  and  therefore  its  motion  in  space  being  compounded 
of  these,  it  moves  as  if  all  the  impulsions  distributed  through  the 
system  were  directly  applied  to  the  centre  of  gravity  or  to  the  whole 
mass  concentrated  there. 

Let  us  now  suppose  the  bodies  in  the  system  to  be  invariably 
connected  together,  as  in  the  case  of  one  solid  mass. 

Let  P,  P',  &c.  represent  the  impulsive  forces  which  act  on  the  se- 
veral bodies,  decompose  each  force  into  two  others  ;  the  one  due  to 
the  motion  which  it  actually  produces,  and  the  other  due  to  the  mo- 
tion destroyed  on  account  of  the  mutual  connexion  of  the  bodies  in 
the  system  ;  so  that  F,  F',  &c.  may  be  the  forces  which  produce 
their  full  effect,  and  ff,  &c.  those  which  are  destroyed  by  the  mu- 
tual action  of  the  parts  of  the  system  ;  thus  F  and/  will  be  the  com- 
ponents of  P  ;  F',  andy"'  the  components  of  P' ;  &c.  In  virtue  of 
the  forces  F,  F',  &c.,  which  are  fully  effective,  the  motion  must  be 
the  same  as  if  these  forces  only  acted  on  the  system,  all  connexion 
between  its  parts  being  destroyed,  so  that  from  what  has  been 
proved  above,  the  centre  of  gravity  ought  to  move  as  if  all  the  forces 
F,  F',  Sec.  were  immediately  applied  to  it,  in  directions  parallel  to 
those  which  they  actually  take.  As  to  the  forces  f,f,  &c.  they  are 
mutually  destroyed  when  they  act  upon  the  several  parts  of  the  sys- 
tem, and  consequently  satisfy  the  six  equations  (6),  (.7),  page  79; 

26 


202  ELEMENTS  OF  DYNAMICS. 

but  when  transported,  parallel  to  themselves,  to  the  centre  of  gravity, 
they  ought,  for  much  greater  reason,  to  be  mutually  destroyed,  since 
then  the  equations  (4)  page  28,  suflice  to  establish  their  equilibrium. 
Hence  the  centre  of  gravity  moves  as  if  the  several  impulsions 
were  immediately  applied  to  it. 

(163.)  Let  us  now  examine  the  motion  of  a  body  which  receives 
an  impulsion  that  does  not  pass  through  the  centre  of  gravity  ;  the 
motion  of  translation  would,  we  know,  from  what  is  proved  above, 
be  the  same  as  if  the  impulsion  were  applied  in  a  parallel  direction 
to  the  centre  ;  but,  beside  this,  there  would  be  impressed  a  motion 
of  rotation,  precisely  the  same  as  would  have  been  impressed  by 
the  same  force  if  a  fixed  axis  had  passed  through  the  centre  of  gra- 
vity. This  double  motion,  arising  from  a  single  impulsion,  may  be  at 
once  shown  to  take  place  as  follows.  Let  P  (fig.  117)  represent  the 
impulsive  force,  and,  perpendicular  to  its  direction,  draw  GA  from  the 
centre  of  gravity ;  at  an  equal  distance  GB,  on  the  other  side  of  G,  let 
two  forces  iP  and  IP,  equal  and  opposite  to  each  other  be  applied, 
these  will  have  no  effect  on  the  system,  so  that  we  may  consider  the 
motion  which  the  body  actually  takes,  in  consequence  of  the  single 
force  P,  to  be  the  result  of  the  three  forces  acting  as  in  the  figure  at  A 
and  B.  The  motion  of  translation  is  due  to  the  force  P,  considered  as 
acting  at  G,  or,  which  is  the  same  thing,  this  motion  is  due  to  the 
force  5  P  acting  at  A,  and  to  d  P  acting  at  B,  in  direction  BS  ;  the 
remaining  forces,  therefore,  that  is  the  force  |P  at  A,  and  the  oppo- 
site force  5  P  at  B,  in  direction  BR,  are  those  to  which  the  rotatory 
motion  is  due  ;  the  tendency  of  these  forces  is  to  turn  the  body  about 
G,  being  symmetrically  situated  with  respect  to  it,  and  the  value  of 
the  forces  to  produce  this  effect  is  at  A,  GAXjP,  and  at  B,  GBxiP, 
and  as  these  forces  turn  the  system  in  the  same  direction  their  whole 
effect  is 

GAx|P+GBxiP=GAxP; 

which  is  the  effect  due  to  the  impressed  force  P  to  turn  the  body 
about  a  fixed  axis  through  s. 

It  follows,  therefore,  that  ?W/m  a  body  is  acted  upon  by  anyim- 
pidsive  forces,  of  ivhich  the  resultant  does  not  pass  through  the  cen- 
tre of  gravity,  the  body  tcill  have,  in  consequence,  a  double  mo- 
tion; 1,  the  centre  will  move  as  if  the  forces  were  immediately  ap- 
plied to  it ;  2,  the  body  ivill  turn  as  if  this  centre  were  absolutely 
fixed. 

Let  P  (fig.  118)  be  the  momentum,  or  quantity  of  motion,  im- 
pressed on  the  body,  r  its  distance  OG  from  the  centre  of  gravity 
of  the  body  M  ;  then,  for  the  velocity  of  translation  due  to  this  force, 

P 

we  have  w=jrj:.     Again,  for  the  angular  velocity  ca  due  to  the  same 


ROTATION    OF    A    SOLID    BODY.  203 

force  P,  acting  at  the  distance  GO  from  the  centre  of  motion,  we 

Pr 

have  (157)  io=^rj7^;  consequently,  the   absolute  velocity  of  any 

point  in  the  body  is  compounded  of  these  two,  viz. 

progressive  velocity,  ^=^  1 

P     r      vr    r (^)' 

angular  velocity,  «=^ .  p=p  J 

r  being  the  perpendicular  distance  of  the  centre  of  gravity  of  the 
mass  M,  from  the  direction  of  the  applied  force  P,  and  k  being  the 
principal  radius  of  gyration.  These  results  may  be  expressed  in 
words,  as  follows,  viz,  the  progressive  velocity  is  equal  to  the  mov- 
ing force,  divided  by  the  mass  of  the  body,  and  the  angular  velo- 
city is  equal  to  the  m,oment  of  the  force  divided  by  the  moment  of 
inertia. 

If,  however,  the  body  is  not  free  to  revolve  about  its  centre  of  gra- 
vity, but  is  constrained  to  turn  about  some  other  point  moving  uni- 
formly, the  angular  velocity  will  be  different.  It  will  be  easy,  how- 
ever, to  estimate  it,  for  as  the  tendency  to  turn  about  the  centre  of 
gravity  is  the  same  on  whichever  side  of  it  the  impulsion  be  given, 
provided  only  it  act  at  the  same  distance  from  it  and  in  contrary  di- 
rections, we  may  obviously  consider  the  angular  motion  which 
accompanies  the  progressive  motion  of  the  point,  to  be  the  result  of 
a  force  acting  in  a  direction  opposed  to  the  progressive  motion,  and 
at  the  opposite  side  of  the  centre  of  gravity,  but  at  the  same  distance 
from  it  as  the  centre  of  motion.  The  value  of  the  impulse  to  which 
the  motion  of  the  body  is  due,rfwill  be  known  from  knowing  the 
progressive  velocity  of  the  centre  of  motion  and  the  mass  moved. 
We  shall  give  an  illustrative  example  of  this  hereafter  (at  prob.  IV., 
chap.  VII.);  at  present  we  consider  the  body  as  entirely  free,  in 
which  case  we  observe  the  following  particulars. 

At  the  instant  the  impact  is  given,  the  point  0  departs  in  the  di- 
rection Oh,  its  initial  velocity  being  equal  to  the  sum  of  its  pro- 
gressive and  angular  velocities  as  their  directions  coincide,  and  the 
same  is,  obviously,  the  case  with  any  other  point  in  GO.  With  the 
points  in  GO',  on  the  other  side  of  G,  it  is  different,  for  although 
they  have  the  same  angular  velocities  as  the  corresponding  points 
in  GO,  yet,  turning  in  the  contrary  direction,  their  absolute  velo- 
city of  any  one  of  them  is  equal  to  the  difference  between  its  pro- 
gressive and  angular  velocities;  that  is  to  say,  every  point  O'  in 
the  line  GO'  has,  in  virtue  of  the  angular  motion  of  the  system,  a 
velocity  backward,  and,  in  virtue  of  the  progressive  motion  of  the 
system,  a  velocity  forward ;  this  latter  is  the  same  for  all  the  points 
in  the  body,  and  equal  to  that  of  the  centre  of  gravity,  but  the 


204  ELEMENTS    OF    DYNAMICS. 

backward  velocity  of  every  point  in  GO'  varies  witli  its  distance 
from  G,  being  0  at  G,  and  increasing  regularly  as  the  distance  in- 
creases; there  must  in  consequence  be  some  point  in  GO',  either 
within  or  without  the  body  of  which  the  velocity  backward  is  pre- 
cisely equal  to  the  velocity  forwards  ;  this  point  then  is  for  an  in- 
stant at  rest  while  all  the  other  points  of  the  body  are  in  motion ; 
so  that  the  whole  system,  when  the  initial  motion  is  given,  turns 
spontaneously  round  it ;  this  point  is  hence  called  the  centre  of 
spontaneous  rotation.  Its  situation  is  readily  determined  from  the 
condition  which  characterizes  it,  which  is,  that  calling  its  dis- 
tance  from   G,  r ',  and   progressive   velocity   of    the   system   v, 

V  fc^ 

V — r'td=0  .*.  r'=— ,  or,  from  equation  (1)  above,  r=— ;  hence,  if 

w  ^  \    y  J. 

0  (fig.  119)  be  the  point  thus  determined,  its  property,  with  respect 

to   the  point  of  impact  0,   is  that  OC  =  r-| —  ....  (2) ;    which 

proves  (159)  that  the  centre  of  spontaneous  rotation  coi)icides  with 
the  centre  of  suspension  corresponding  to  the  point  of  percussion, 
considered  as  the  centre  of  oscillation,  and  is  entirely  independent 
of  the  intensity  of  the  applied  force. 

(164.)  In  order  to  determine  at  what  distance  GO,  from  the 
centre  of  gravity,  the  impulsion  must  have  been  given  to  produce 
the  actual  progressive  and  rotatory  motions  observed  in  any  body, 

we  have,  from  the  equation  (1)  above,  r= ;  or,  if  V  be  the  ro- 

V 

tatory  velocity  of  any  point  at  the  distance  R  from  the  centre  of 

V  .        k'   V 

gravity,  then  since  "=p-»  we  have  r=-^.  — .  (1). 

Applying  this  to  the  double  motions  of  the  planets,  we  may  deter- 
mine at  what  distance,  from  the  centre  of  each,  the  original  impul- 
sion must  have  been  impressed  by  the  hand  of  the  Creator  to 
cause  their  actual  motions  of  progression  in  space  and  rotation  on 
their  axes. 

Taking  the  earth  for  example,  we  know  that  it  performs  its  revo- 
lution on  its  axis  in  a  sidereal  day,  by  which  rotatory  motion  every 
point  on  the  equator  passes  over  about  25020  miles. 

Also  its  orbit,  or  a  path  of  about  596904000  miles,  is  passed 
over  by  its  progressive  motion  in  366  sidereal  days,  hence  the  ratio 

V.   ,  V      596904000       ^     „ 

— isherel  -. — = =  6o.  3  ; 

V  V     25020x366 

and,  considering  the  earth  a  sphere,  we  have  (p.  189)  A:*=|R"; 
hence,  by  substituting  these  values  in  tlie  formula  (1),  we  have  for 
the  distance  r  from  the  centre  of  tlie  sphere  at  which  the  impulse 


ILLUSTRATIVE  PROBLEMS.  205 

T?  1 

was  given  r  :=  — - — — ;  that  is,  about  the  -— —   part  of  the   radius 
Loo  '  Zi  lt)o 

distant  from  the  centre. 

It  is  very  probable  that  not  only  the  planets  but  that  also  the 
sun  may  thus  derive  its  motion  from  a  single  primitive  impulse,  and 
if  so,  he,  in  common  with  the  planets,  must  also  have  a  progress- 
ive motion  in  space  ;  this  cannot,  indeed,  be  rigorously  proved. 
"  But,"  to  use  the  words  of  Dr.  Robison,  as  quoted  by  Professor 
Gregory,  "  the  very  circumstance  of  his  having  a  rotation  in  27d. 
7h.  47m.  makes  it  very  probable  that  he,  with  all  his  attending 
planets,  is  also  moving  forward  in  the  celestial  spaces,  perhaps 
round  some  centre  of  still  more  general  and  extensive  gravitation  : 
for  the  perfect  opposition  and  equality  of  two  forces  necessary  for 
giving  a  rotation  without  a  progressive  motion,  has  odds  against  it 
of  infinity  to  unity.*  This  corroborates  the  conjectures  of  philo- 
sophers, and  the  observations  of  Herschel  and  other  astronomers 
who  think  that  the  solar  system  is  approaching  to  that  quarter  of 
the  heavens  in  which  the  constellation  Aquila  is  situated." 


CHAPTER  IV. 

PROBLEMS  ILLUSTRATIVE  OF  THE  PRECEDING  THEORY. 

(165.)  We  shall  now  proceed  to  illustrate  the  theory  delivered 
in  the  two  preceding  chapters,  by  showing  its  practical  application 
in  a  few  miscellaneous  problems.  Several  of  these  are  those  se- 
lected by  Mr.  Barlow  in  his  Treatise  on  Mechanics,  in  the  Ency- 
clopedia Metropolitana. 

Problem  I. — Let  AB  (fig.  120)  denote  an  axle  turning  on  two 
fulcrums  at  A  and  B ;  RS  a  wheel  of  given  diameter  to  which  is 
attached,  by  a  cord  wound  round  its  circumference,  a  given  weight 
AV;  conceive  ;>,p',  ^>", /j'",  to  be  given  weights,  fixed  to  the  axes 
by  inflexible  lines  or  wires,  Cjo,  CjO',  &c.  and  let  it  be  required  to 
determine  the  circumstances  of  the  motion  of  the  descending  weight. 

We  shall,  in  the  first  place,  consider  this  problem  under  its  most 
simple  form,  viz.  we  shall  suppose  the  wheel  RS,  the  axle  AB,  and 

*  It  does  not  appear  to  us,  however,  tLat  any  weight  should  be  attached  to 
this  assertion,  founded  on  the  doctrine  of  chances,  and  which  can  strictly  apply 
only  to  the  case  of  two  impulsive  forces,  directed  at  random  towards  opposite 
parts  of  a  spherical  body.  Whatever  primitive  motions  the  Almighty  may  have 
designed  to  impress  on  the  sun,  the  impulses  it  must  have  received  could  not  fail 
to  be  those  precisely  competent  to  produce  the  intended  effect. 

s 


206  ELEiMENTS   OF    DYNAMICS. 

the  lines  or  wires  Cp,  Cp',  &c.  as  divested  of  inertia,  qv  as  offer- 
ing no  resistances  to  angular  motion ;  so  that  W  and  the  four 
masses  p,  p',  p",  p'",  will  be  the  only  weights  in  the  system,  the 
latter  being  all  equal,  placed  at  equal  distances  from  the  axle  C, 
and  at  right  angles  to  each  other. 

Let  Cp,  Cp'  &c.=:r;  the  radius  of  the  wheel  =r' ;  the  sum  p 
+/)'  +  p"H-/)"'=P,  and  the  given  weight  =W;  then  the  moment 

P 
of  inertia  of  P  will  be  r^ — ,  and  the   moment  of  W,  the  moving 

force,  will  be  r'W,  consequently  the  acceleration  of  W,  being  equal 
to  r'  times  the  angular  acceleration,  will  be 

W         P  r'  ^W 

r'^W-j-  r'* 1-  r^—  that  is,  F=     ,,-. rr  sr, ;  and  the  velocity  of 

g  g  r'^W+r^P®  -^ 

W,  after  any  time  t",  will  be 

r'nv 

D^  F/  = erf  • 

and  the  corresponding  space  descended  will  be 

and  thus  the  circumstances  of  Ws  motion  are  all  known. 

Let  us  now  take  into  consideration  the  inertia  of  the  wheel  and 
axle;  call  the  weight  of  the  former  W'  and  that  of  the  latter  W", 
also  let  r"  be  the  radius  of  the  axle;  then  (page  187,) 

VV  W" 

the  moment  of  inertia  of  the  wheel  =  5  r'*  — ,axle=  5  r"* , 

•  .  .      S  ,  S 

the  square  of  the  radius  of  gyration  being  in  both  cases  A:*=2r*. 

The  motive  force  being  still  the  same  as  before,  we  have,  for  the 
acceleration  of  W, 

p r^^ 

r'MV4-dr'2  W'  +  |r"2  VV"+r-P^' 

The  accelerating  force  being  thus  known,  the  space,  velocity,  time , 
«fcc.  are  determined  by  the  usual  formulas  for  constant  forces. 

If  we  suppose  the  system  of  small  weights/?,  p',  p",  &,c.  to  be 
replaced  by  a  solid  body  of  revolution,  as  in  fig.  (121)  the  principles 
of  the  calculation  will  be  still  the  same ;  for  the  moment  of  inertia 

P  . 

of  the  solid  P  will  as  before  be  A:'—,  k  being  the  principal  radius  of 

gyration  as  measured  from  the  geometrical  axis.  Thus  in  fig,  121, 
let  P  denote  a  sphere  whose  radius  is  3  feet,  and  weight  500  lbs.  ; 
the  weight  W:=50  lbs.,  and  the  radius  of  the  wheel  6  inches,  or  |  a 
foot,  and  of  which  the  weight,  as  well  as  that  of  the  axle,  are  sup- 
posed inconsiderable  with  respect  to  the  other  parts  of  the  system  ; 


](U     / 

,s 

\ 

Ix'' 

" -•. 

"    --■ 

, 

ILLUSTRATIVE  PROBLEMS.  207 

and  let  it  be  required  to  determine  the  time  in  which  the  weight  W 
will  pass  through  any  given  space,  as,  for  example,  50  feet. 

In  the  sphere  k^=fr'';  hence  the  expression  for  the  acceleration 
of  Wis 

p_ r_-W 4^X50  _  402^V 

,.'aW+|r^P   ^       Ix50  +  fx9x500 '        18121 

hence  the  time  of  descent  is  15". 


Problem  II. — Let  ABC  (fig.  121)  represent  a  wheel  and  axle,  its 
weight  w,  having  a  given  weight  W  applied  to  the  circumference  of 
the  axle,  and  P  applied  to  the  circumference  of  the  wheel  in  order 
to  raise  W ;  it  is  required  to  assign  the  space  described  by  the  ele- 
vated weight  W  from  rest,  in  any  given  time. 

Let  the  radius  of  the  axle  be  r,  that  of  the  wheel  R,  and  the  prin- 
cipal radius  of  gyration  of  the  wheel  k ;  then,  for  the  moment  of 
inertia  of  the  whole  system,  we  shall  have  the  expression 
,   w      ^P  W 

^3  _  ^   R2_  +   ,.3  _  . 
or  or  or 

O  o  _      S 

Now  the  actual  weight,  which,  applied  at  the  point  D,  gives  mo- 
tion to  this,  is  not  the  whole  weight  P,  since  part  of  this  is  employ- 
ed in  balancing  the  weight  W ;  to  know  what  this  part  is,  we  have, 

Wr 
by  calling  it  P',  the  equation  Wr=P'R  .-.  P'= — p—  so  that  only 

the  weight  P ~  = 

is  actually  employed  in  moving  the  system,  and  as  this  weight  acts 
at  the  point  D  we  must  multiply  it  by  R  to  obtain  its  moment,  and 
dividing  R  times  this  by  the  mass  moved,  or  by  the  whole  inertia 
we  have,  for  the  acceleration  of  P,  the  expression 
PR2_WRr 
k^w+n^P  +  r^W    ^  •  •  '  '^  >• 
Now  as  the  acceleration  of  P  is  to  the  acceleration  of  W  as  the  ra- 
dius R  to  the  radius  r,  we  have 
R  .     •  •         PR^  — WRr  PRr  —  Wr''         ^ 

which  expresses  the  acceleration-  of  the  ascending  weight. 
If  R=r  the  acceleration  of  either  weight  will  be 

p W 

F= — - R2  2- 

A;^w;-fR2(P-|-W)       ^* 

It  should  be  remarked,  that  if  the  mass  moved,  W,  have  no  weight 


208  ELEMENTS    OF    DYNAMICS. 

but  inertia  only,  or  rather  if  its  weight  is  otherwise  supported,  and 
its  inertia  only  has  to  be  overcome  by  tlie  ni:'.chine,  as  for  instance 
when  it  is  to  be  moved  along  a  perfectly  smooth  horizontal  plane, 
then,  in  the  niinierators  of  the  foregoing  expressions,  we  must  put 

■w=o. 

Problem  III. — Let  ABC  (fig.  121,)  represent  a  wheel  and  axle 
of  given  weight  moveable  about  a  horizontal  axis  which  passes 
through  S ;  and  suppose  a  known  weight  W  is  applied  to  the  cir- 
cumference of  the  axle,  to  be  raised  by  a  given  force  P  applied  to 
the  circumference  of  the  wheel ;  to  assign  tlie  proportion  of  the 
radii  of  the  wheel  and  axle,  so  that  the  time  in  which  tlie  weight  W 
ascends  through  any  given  space  shall  be  a  minimum. 

Since  the  ratio  only  of  the  radii  of  the  wlieel  and  axle  is  required, 
let  the  radius  of  the  axle  be  r,  and  that  of  the  wheel  xr  ;  the  weight 
of  the  wheel  iv  and  k  as  before,  the  principal  radius  of  gyration  of 
the  wheel,  and  of  which  the  value  we  know  is  (p.  188)  k"=i  x^r". 
Then,  substituting  xr  for  R  and  ^x^r''  for  A:*,  in  equation  (2) 
of  the  preceding  problem,  we  have,  for  the  acceleration  of  W,  the 

I»  X W 

expression  -— ; — — ——{^  g  ^  and  consequently 

{iw+i\)x^-]-w^    "      ^'     ig(yx—\y)    ^' 

This  expression  is  to  be  a  minimum,  and  consequently  the  quantity 
under  the  radical  will  be  a  minimum  ;  therefore,  dividing  this  by 

o 

the  constant  7—,  and  putting  for  brevity  J9  for  ^  U)-\-V,  we  have 

■is 

V; ^r-.  =  a  minimum ; 

hence,  by  difTerentiating, 

2px{Px—W)—V{px^-^W)=0  .'.  Fpx"  —  2Wpx=V\\ 

w  ^\^   w , 

If  the  weight  of  the  wheel  be  too  inconsiderable  to  deserve  notice, 

Wrb  x/^W'^  +  PW^ 

then  jO=P,  and  in  this  case  x= '— (2),  and  if, 

moreover,  P=W,  we  have  .r=l  zb  ■v/2. 

Suppose,  for  example,  ABC  to  represent  a  cylindrical  wheel,  the 
radius  of  which  is  required,  but  of  which  the  weight  is  20  lb ;  and 
let  the  radius  of  the  axle  be  1  inch  ;  the  weight,  W,  100  lb.,  and  the 
weight  P,  33  lb.  ;  to  find  the  radius  of  the  wheel. 

Here  5  w-\-F=p=4S\h.,  therefore,  by  equation  (1), 

100         ,  100^      100' 


ILLUSTRATIVE    PROBLEMS.  209 

Consequently,  since  the  radius  of  the  axle  is  1  inch,  the  radius  of 
the  wheel  must  be  6*43  inches. 

For  other  such  problems  as  this,  the  student  may  consult  Dr. 
Gregory's  chapter  on  the  "  Maximum  Effects  of  Machines." 

Problem  IV. — In  the  wheel  and  axle,  when  a  given  weight  P 
acting  at  the  distance  R  raises  a  weight  W  acting  at  the  distance  r 
from  the  geometrical  axis,  it  is  required  to  assign  the  pressure  sus- 
tained by  the  axis,  the  weight  of  the  wheel  and  axle,  and  the  fric- 
tion of  the  cord  not  being  considered. 

Suppose  P  and  W  to  be  at  their  respective  extremities  of  the  ho- 
rizontal line  passing  through  the  centre  of  motion,  and  in  that  situa- 
tion let  O  and  G  be  the  respective  distances  of  the  centres  of  oscil- 
lation and  gravity  from  the  centre  of  motion  ;  then,  since  the  angu- 
lar velocity  of  any  revolving  system  is  the  same  as  if  its  whole  mass 
were  concentrated  in  its  centre  of  oscillation,  we  may  consider 
P  +  W  to  be  placed  at  the  distance  O  from  the  centre  of  motion  ; 
and,  since  the  force  of  any  point  in  the  revolving  system  is  propor- 
tional to  its  distance  from  the  axis  of  motion,  we  have 

0:G  ::P-f  W:^(P  +  W), 

which  is  the  force  or  pressure  with  which  the  centre  of  gravity  de- 
scends ;  or  in  fact  the  force  with  which  the  whole  mass  descends. 
Part  of  the  whole  pressure  P  +  W  of  the  system  is  thus  supported 

C 

by  the  axle,  and  the  other  part,  which  we  have  just^  seen   is  — 

X(P-|-W),  is  employed  in  producing  the  motion  which  actually  has 
place  ;  consequently,  that  part  of  the  pressure  sustained  by  the  axle 
must  be 

P+W-^(P+W). 

It  remains,  therefore,  to  find  the  values  of  G  and  O,  which  are 
_  PR  — Wr  _  PR^-fWr" 

-     P  +  W~'        ~"   PR_Wr  ' 

hence,  by  substitution,  we  have  for  the  pressure  j), 

'  ^  PR^  +  Wr^  PR^-fWr^^ 

If  R  =  r,  as   in  the   case   of  the   single   fixed  pulley,   then   the 

4PW 
pressure  is  p  =  p  ^^. 

Problem  V. — Let  A,  B  (fig.  122,)  represent  a  single  moveable 
pulley  by  means  of  which  the  power  P  elevates  the  weight  W; 
s2  27 


210  ELEMENTS  OF  DYNAMICS. 

then,  having  given  P  and  W,  together  witli  the  weights  of  the  equal 
cylindric  pulleys  A  and  B,  it  is  required  to  assign  the  space  which 
the  descending  weight  P  describes  in  a  given  time,  the  weight  of 
the  moveable  pulley  being  included  in  the  weight  W. 

Let  us  refer  the  whole  inertia  of  the  system  to  the  point  p,  so 
that  we  may  consider  tlie  force  which  moves  P  to  be  burdened  with 
the  mass  of  P,  and  with  the  additional  mass  representing  the  inertia 
of  the  other  parts  of  the  system,  this  mass  being  all  accumulated 
at  /;,  or,  which  is  the  same  thing,  incorporated  in  P. 

In  the  first  place  the  inertia  of  the  pulley  A,  whose  weight  call 
Q,  is  the  same  as  that  of  half  its  mass  placed  at/j,  (see  page  188.) 

In  like  manner  the  inertia  of  the  pulley  B  is  the  same  as  that  of 
half  its  mass  placed  at  q ;  or,  since  the  rotatory  velocity  of  B's 
circumference  is  only  half  the  velocity  of  A's  circumference,  the 
mass  of  half  B  at  y  has  the  same  inertia  as  the  same  mass  placed 

at  half  the  distance  Op,  or  finally  as  J    times  the  same   mass 

placed  atj3,  (p.  188.)    Hence,  as  far  as  the  inertia  of  the  pulleys  is 

Q         Q 

concerned,  the  equivalent  mass  to  be  placed  at/)  is  5  . f- ^.— . 

Again,  as  the  velocity  of  W  is  half  that  of  P,  it  move's  as  if  it 
hung  at  half  the  distance  Op'  from  the  centre  of  the  pulley  A, 
where,  as  in  the  case  of  the  pulley  B,  it  would  offer  the  same  in- 
ertia as  ^  its  mass  placed  at  p'  or  p ;  hence  the  inertia  of  the  weight 

P       W 

is  represented  by  the  mass 1-  —  placed  at  p,  so  that  the  whole 

mass  moved  by  the  force  which  moves  P  is 

P  +  ^W-f  IQ+jQ  _  SP-f  2W-f5Q 

g  ^g  '     . 

To  determine  now  the  force  or  weight  which  moves  this  mass, 
we  must  find  how  much  of  the  applied  weight  P  is  employed  in 
merely  balancing  W ;  this  is  easily  done,  because,  as  W  is  equally 
supported  by  the  two  branches  of  rope/)'^,  Oq' ,  one  half  of  W  is  the 
portion  of  P  employed  in  balancing  W ;  hence  the  moving  force  is 

and  consequently  for  the  acceleration  of  P  we  have 

2P_W       8P+2W-f5Q  8P  — 4W 


F  = 


2  ■  Sg  8P  +  2VV  +  5Q' 

8P_4W  8P  — 4W         ff/» 

^gt^s- 


8P+2W+5Q*'        8P  +  2W-f5Q     2 
Problem  VI. — In  a  system  of  pulleys  contained  in  two  equal  and 


ILLUSTRATIVE    PROBLEMS.  211 

separate  blocks  a  single  string  goes  round  them  all :  it  is  required 
to  determine  the  acceleration  of  P  fastened  at  the  extremity  of  the 
string,  and  drawing  up  a  weight  attached  to  the  lower  block. 

Let  there  be  altogether  n  pullies,  Q  being  the  weight  of  each, 
and  let  the  weight  of  the  lower  block,  together  with  its  attached 
load,  be  W,  and  let  us,  as  in  the  preceding  problem,  ascertain  what 
mass  in  addition  to  its  own  must  be  incorporated  in  P  to  supply  the 
place  of  the  inertia  of  the  system. 

The  pulley  which  first  received   the  cord  from  the    ascending 

weight  turns  with  — th  of  the  velocity  of  the  pulley  which  delivers 

the   cord  to  the  power,  and  therefore,  as  in  the  last  problem,  the 

1      Q 

mass  at  P,  which  will  be  a  substitute  for  its  inertia,  is .  — . 

2  .  /i^     g 

The  circumference  of  the  next  pulley  in  the  lower  block  revolving 

twice 'as  fast  as  the  foririer,  the  mass  due  to  its  inertia  will  be 

4       Q 

- — •  .  — ,  and  so  on ;  hence,  for  the  pulleys,  the  mass  to  be  substi- 
2.«^     g  '        ■' 

tuted  will  be 

2.nr  g  ^  '       2n^  g  6 

Again,  the  velocity  of  W  being  — th  of  P's  velocity,  the  mass  at 

P,  which  represents  its  inertia,  will,  as  in  the  preceding  problem,  be 

1  W 

— . — ;  hence  the  inertia  of  the  weights  is  represented  by  the  mass 

P      1    W 

— I .  — ,  and  therefore  the  whole  mass  moved  is 

g      n^    g 

—  4.—    ^^  \       ^     ^  2  n''  +  3  n  +  1 
g        n^'  g       2ng   '  6  ^ ' 

W 

The  weight  at  P  which  balances  W  is  —  and   consequently  that 

n, 

W      n  P W 

which  moves  the  system  is  P = ,  and  therefore,  di- 

n  n 

viding  this  by  the  mass  moved,  as  above  expressed,  we  have  for 

^T,     T.  12wrnP— W)^ 

the  acceleration  of  P  ;  F  ==  — 7 — rr ,.,,,       ^ — ttt^- — -;: 7t> 

12  (nM^  +  W)  +  Qn  (2  n«  +  3  n  +  1) 

from  which,  as  in  the  preceding  problem,  the  expressions  for  the 

velocity  and  space  generated  in  any  time  t"  are  immediately  dedu- 

cible. 

Problem  VII. — A  wheel,  whose  interior  and  exteror  radii  are 


212  ELEMENTS  OF  DYNAMICS. 

fj,  r^,  rolls  down  an  inclined  plane  (fig.  123),  of  which  the  friction 
is  just  sufficient  to  prevent  sliding  :  to  determine  the  circumstances 
of  the  motion. 

Let  i  be  the  inclination  of  the  plane,  and  F  the  effective  accele- 
rative  force  down  it,  then,  putting  tv  for  the  mass  of  the  wheel,  Fw 
will  represent  the  efFective  moving  force.  But  the  impressed  mov- 
ing force  is  ivs^  sin.  i,  minus  the  resistance  of  the  friction,  which, 
as  it  diminishes  the  amount  of  the  moving  force  which  the  body 
would  otherwise  have,  we  may  represent  by  an  opposing  moving 
force ;  let  us  then  call  it  w'g  sin.  i.  Now  if  P  be  the  point  in  con- 
tact with  the  plane  at  D,  where  the  motion  is  supposed  to  have  com- 
menced, then,  since  in  any  time  t",  DP'  =  P'P,  it  follows  that  the 
rotatory  acceleration  of  P  is  also  F ;  and,  consequently,  the  accele- 
ration of  a  particle  at  the  unit  of  distance  from  C,  that  is  the  angu- 

F 

lar  acceleration,  is  — ;  hence  wk'  being  the  moment  of  inertia  of 

F 

the  wheel  round  C,  —  ivk''  will  be  the  actual  moment  of  the  sys- 

tem  round  C.    But  the  only  impressed  moment  is  that  arising  from 

the  friction,  it  is,  therefore,  r^  w'g  sin.  i  ;  hence,  by  the  principle 

of  D'Alembert,  we  have,  by  equating  the  impressed  and  effective 

forces, 

r.        ,  ,^      .      .    Fwk^ 

Fw={w — wjgsin.t;   =r^w  gsm.i; 

eliminating  w',  we  have  Fw  r^'^-^-Fw  k^=wrj^^  g  sin.  t 
.  p^g  sin,  i  r^" 

which  expresses  the  acceleration  of  the  centre  C  down  the  plane, 

r  *-4-r  ^ 
and  in  which  k'>=-!--~-,  (ex.  3,  page  188.) 

If  the  cylindrical  wheel  be  indefinitely  thin,  or  when  r^=r^, 
F=5  g  sin.  i. 
For  the  time  of  describing  a  given  space  s,  we  have, 

g  sm.  t  r^  ' 

2  5  '^  sin.  X  T  ^ 
and  for  the  velocity  acquired,  v^=Ft=^  \ — "^     '     ^  \,  which 

expresses  also  the  rotatory  velocity  of  every  point  on  the  outer  cir- 
cumference. 

To  determine  the  absolute  velocity  of  any  proposed  point  P  of 
the  outer  circumference  in  space,  or  in  its  cycloidal  path,  we  must 
compound  together  these  two  velocities ;  so  that,  calling  the  abso- 
lute velocity  V,  we  have 


ILLUSTRATIVE  PROBLEMS.  213 

V=2  V  cos,  5  w  P  m=2  v  x  tab.  cos.  |  arc.  P^;  m  ; 
and  the  degrees  in  Vpnl  are  known,  since  the  length  2  s  of  the  arc 
PP'm  is  known ;  therefore  the  absolute  motion  of  P  is  determined 
both  as  to  velocity  and  direction. 

\i  p  be  diametrically  opposite  to  the  point  of  contact  P',  it  ap- 
pears from  this  expression  for  V,  that  the  absolute  velocity  of  znf 
point  P  at  any  time  varies  as  the  tabular  cosine  of  half  the  arc  Vp ; 
at/3  this  velocity  is  gi-eatest,  being  =:  2v  ;  and  at  P'  it  is  least, 
being  for  an  instant  0,  that  is,  P'  is  the  centre  of  spontaneous  rota- 
tion of  the  body.  The  last  conclusion,  viz.  that  the  point  of.  con- 
tact can  have  no  motion  along  the  plane,  is  an  immediate  conse- 
quence of  the  conditions  of  the  problem,  for  if  it  had  any  motion 
along  the  plane,  the  body  would  slide  on  that  point,  whereas  the 
friction  is  supposed  to  be  sufficiently  gi-eat  to  prevent  sliding  down 
the  plane. 

If,  in  consequence  of  any  initial  impulse,  the  rotatory  velocity 
exceed  that  of  translation,  P'  will  no  longer  be  the  centre  of  spon- 
taneous rotation,  but  will  have  a  velocity  backward  greater  than  that 
forward,  so  that  the  body  will  move  up  the  plane  and  will  continue 
to  do  so  till  this  excess  of  rotatory  velocity  is  destroyed  by  the  fric-- 
tion ;  when  the  body,  after  being  for  an  instant  stationary,  will  re- 
verse its  progressive  motion,  and  roll  down  the  plane  with  a  velo- 
city equal  to  that  of  rotation.  If  the  velocity  of  translation  were 
made  to  exceed  that  of  rotation  the  body  would  partly  roll  down 
the  plane  and  partly  slide.  These  deductions  are  on  the  supposi- 
tion that  the  friction  of  the  plane  is  just  sufficient  to  prevent  sliding 
when  the  initial  velocity  of  the  body  in  progression  is  equal  to  that 
in  rotation. 

Rotatory  motion  of  this  kind  may  be  produced  by  the  uncoiling 
of  a  thread  or  riband  previously  w'ound  about  the  body ;  the  ten- 
sion of  the  thread  supplying  the  place  of  friction. 

Problem  VIII. — A  sphere,  whose  mass  is  P,  is  placed  on  the 
slant  side  of  a  smooth  prism,  whose  mass  is  Q,  and  a  fine  thread  or 
riband  is  fixed  to  the  prism  at  B  (fig.  124,)  and  coiled  round  the 
sphere  in  the  plane  of  its  vertical  gi-eat  circle,  the  object  of  it  being 
to  cause  the  sphere  to  roll  and  not  slide  down  the  plane.  The  base 
AC  of  the  prism,  as  well  as  the  horizontal  plane  on  which  it  is 
placed,  is  perfectly  smooth,  so  that  it  moves  along  the  plane  OC, 
in  consequence  of  the  pressure  of  the  rolling  sphere.  It  is  required 
to  determine  the  tension  of  the  string,  at  any  time  the  pressure  on 
the  prism,  and  the  path  described  by  the  point  of  contact  P. 

In  this  case  the  rolling  body  has  two  motions,  viz.  one  down  the 
inclined  plane,  and  the  other  in  a  horizontal  direction,  as  well  as 
the  rotatory  motion  about  iis  centre.     As  in  last  problem,  the  rota- 


214  ELEMENTS    OF    DYNAMICS. 

tory  acceleration  of  any  point  in  tlie  circumference,  must  be  equal 
to  the  acceleration  of  the  point  of  contact  P  down  the  plane ;  but 
this  rotatory  acceleration  is  wholly  produced  by  tlic  tension  T  of 
the  thread,  it  will,  therefore,  be  expressed  by  dividino^  this  force  by 
the  moment  of  inertia  of  the  sphere,  that  is,  the  acceleration  of  the 
point  of  contact  is  T-^fP,(p.  190-1.)  Let  us  now  deduce  another 
expression  for  this  acceleration  of  the  point  P  from  tlie  actual  space 
BP  passed  over  by  it,  and  for  this  purpose  put  ON=x,  OA=.rj,  O 
being  the  place  of  A  at  the  commencement,  or  when  P  is  at  B  ;  let, 
moreover,  BC=o,  AC=6,  AB  =  c;  then  the  space  BP  is 

BP=(Xj+6  —  a:)  sec.  A=(Xj-f  6  —  ^)7:' 

and  therefore  the  second  differential  coefficient,  with  respect  to  the 
time,  must  express  the  acceleration  down  the  plane ;  that  is, 
c  ,d^x.^       d^x       5'V 
J^~dF~~dF^^2¥  ^  ^* 

The  first  of  these  differential  coefficients  denotes  the  acceleration 
of  the  point  A  or  of  the  entire  prism  in  a  horizontal  direction,  and 
the  second  denotes  the  acceleration  of  the  point  P,  or  of  the  sphere 
in  a  horizontal  direction.  Now  the  motive  forces  to  which  these 
accelerations  are  due,  are  equal  and  opposite,  therefore,  calling  p  the 
pressure  on  the  prism,  or — p,  the  resistance  against  the  sphere,  we 
have  for  the  horizontal  force  on  the  sphere 

dt^  '    c         c  ^  ^ 

dt^       ^  c  c  ^  ' 

Also,  for  the  vertical  force  on  P,  we  have 

dt^ 
Consequently,  since  y  «=  ^{x — x{)  ....  (5), 
the  equations  (1)  and  (4)  give 

Also  the  equations  (1),  (2),  (3),  give 

and  from  these  two  equations  we  obtain  for  p  and  T  the  values 

6cP(2P+7Q)  g 

^^7  a^  (P+ QJ+y^"(2P'+7Q) 


^ey  =  _P,.+^l+T4....(4). 


a 


MOTION    OF    A   SYSTEM    OF    BODIES.  215 

2acF(P+Q)g 
7a^(P  +  Q)+6H2P+7Q) 
both  of  which  are  constant  quantities. 

As  to  the  path  described  by  the  point  of  contact  P,  it  is  imme- 
diately deducible  from  the  conditions  of  the  problem ;  for,  as  both 
bodies  commence  their  motion  together,  the  horizontal  momentum 
of  the  sphere  is  equal  and  opposite  to  that  of  the  prism,  that  is, 

that  is,  (equa.  5,) 

hence  the  path  is  a  determinate  straight  line. 

For  a  very  complete  and  elegant  solution  to  this  problem,  con- 
sidered under  different  modifications,  the  student  may  consult  a 
paper  by  Mr.  Mason,  in  the  twentieth  number  of  Leybourn's  Ma- 
thematical Repository. 


CHAPTER  V. 

ON    THE    MOTION     OF     A   SYSTEM    OF    BODIES    ACTED    ON    BY   ANY 
ACCELERATIVE    FORCES    WHATEVER. 

(166.)  We  propose  in  this  chapter  to  investigate  some  very 
general  and  remarkable  theorems  which  apply  to  the  motion  of  a 
system  of  bodies  acting  in  any  arbitrary  manner  on  each  other, 
and  each  influenced  by  any  accelerative  forces. 

Let  m,  m^,  m^,  &lc.  represent  the  masses  of  the  different  bodies 
in  the  system,  x,  y,  z ;  x^,  y^,  z^,  &c.  the  rectangular  co-ordinates 
which  mark  their  position,  X,  Y,  Z,  the  components  of  the  accele- 
rative forces  on  m,  X^,  Y^,  Zj,  the  components  of  those  on  m^,  &c. 
then  the  motive  forces  applied  to  any  one,  as  m,  will  be  inX,  mY, 

d^  X       d^v 
mZ,  and  those  which  actually  have  place  will  be  m—j-^,  wi-r^, 

(la  z 

m ;  hence  the  differences  of  the  impressed  and  effective  forces 

dt^ 
resolved  in  the  directions  of  the  co-ordinates  are 
d^x  ^       d^y  ^^       d^z 

and,  in  like  manner,  for  each  of  the  other  bodies  m^,  m^,  &c.  we 
get  similar  expressions  for  the  differences  between  the  impressed 


216 


ELEMENTS  OF  DYNAMICS. 


and  effective  forces,  and  we  know,  from  the  principle  of  D'Alembert 
(154),  that  if  these  dilfercnces  alone  acted  on  the  system  it  would 
he  kept  in  equilibrium.  But  when  any  forces  keep  a  system  in 
equilibrium  these  forces  must  fulfil  the  conditions  (6)  and  (7),  at 
page  79 ;  hence,  in  the  present  case,  we  must  have  these  two 
groups  of  equations,  viz. 


2(m-^)=2:(mX) 


(A) 


^      'dV' ^^^  (77?^X  — »nxY) 

,     xd^z — zd^x,         ,      rt  vx 

l{m ^^ )=2  (ma:Z  —  mzx)  L 


S  (w 


dV 
zd^y — yd^z 


(B). 


rf/= 


)==  '2,{mzY  —  myT.)  { 


These  two  groups  of  equations,  which  contain  the  conditions  of 
the  motions  of  any  system  of  bodies  under  any  circumstances,  fur- 
nish several  general  principles  of  motion ;  one  or  two  of  these  we 
shall  proceed  to  develope. 

Let  X,  Y,  z,  represent  the  co-ordinates  of  the  centre  of  gravity  of 
any  system  of  bodies  m,  m^,  m^,  Sic. ;  then  (39)  we  shall  have 
S  (mx)         2  (my)         S  (mz)  _ 


I,{m) 


2  (w) 


2  (in) 


and  taking  the  second  differential  co-efficienls  with  respect  to  t 


d' 


S  (m 


dt''  'd"  z 


,     d'^z 


rf/a        2  (m)      dV'  2  (m)       dt»  2 

Comparing  these  with  the  equations  (A)  we  have 
d"  x_2  (??iX)  rf"  Y     2  (mY)     rf"  z_2  (mZ) 


(m) 


(C); 


dP       2  (m)  '  rf/^        2  (m)  '  rf/2       2  (m) 
or  putting  M  for  the  whole  mass,  m-\-m^-\-m^-\-,  &c.  of  the  system, 
we  have 

M^=2(mX),M^=2(7nY),  M^=2(7nZ); 

These  equations  show  that  if  the  whole  mass  of  the  system  were 
to  be  concentrated  into  the  centre  of  gravity,  and  this  centre  to 
have  the  same  acceleration  as  in  the  actual  state  of  the  system,  the 
moiino^  force  of  the  whole  mass  at  the  centre  would  be  the  same 


MOTION    OF    A    SYSTEM    OF    BODIES.  217 

as  the  entire  moving  force  of  the  actual  system ;  but  S  (?nX)  being 
independent  of  x,  x^,  x„,  &c.  is  independent  of  the  distances  be- 
tween the  several  bodies  m,  m^,  m^,  &c.  and,  therefore,  would 
remain  the  same  if  these  masses  were  united  in  a  single  point, 
provided  only  the  same  accelerative  forces  were  applied  parallel  to 
their  actual  directions ;  hence  the  motion  which  the  centre  of  gra- 
vity actually  has  is  the  same  as  it  would  have  if  all  the  system  were 
united  there,  and  the  forces  applied  to  it  parallel  to  the  directions 
they  really  have ;  whence  this  general  principle  of  motion,  viz. 

T7ie  centre  of  gravity  of  a  system  of  bodies,  acted  upon  by  any 
accelerative  forces  and  mutually  influencing  each  other,  moves  in 
space  as  if  the  system  were  united  into  that  centre,  and  the  forces 
which  solicit  the  bodies  were  directly  applied  to  it. 

(167.)  When  the  system  is  acted  on  by  no  other  forces  than  the 
mutual  attractions  of  its  parts,  the  second  members  of  the  equa- 
tions (A)  must  vanish ;  for,  as  the  action  of  any  two  of  the  bodies 
is  mutual,  each  will  impress  on  the  other  the  same  motive  force ; 
and  as  these  forces  are  opposite  to  each  other,  it  follows  that  if  the 
bodies  were  connected  with  each  other  by  rigid  rods,  these,  on 
account  of  the  sequel  at  opposite  pressures  at  their  extremities, 
would  be  held  in  equilibrium,  whence  all  motion  in  the  system 
would  be  prevented  by  the  forces  applied  to  its  several  parts  thus 
connected  together ;  hence  these  forces  must  fulfil  the  six  equations 
of  equilibrium,  so  that  we  must  here  have 

2  (mX)=0,  2  (mY)=0,  S  (rnZ)=0  ....  (D) 
2  (myX'—mxY)=0,  2  (m^rZ— ?ncX)=0,2  (mzY — ?m/Z)=0 .  (E); 
and,  consequently,  from  the  equations  (C),  we  get 

d^x  d^x  fPz 

———^0,     ,  ^  =0,  -——=0,  of  which  the  integrals  are 
dt^  dt^  dt^  '^ 

x=a-\-bt,  Y=a'-|-67,   z=a"-j-b"t 
where  a,  b,  a',  b'  a",  b",  are  the  arbitrary  constants,  introduced  by 
the  integration. 

If  we  eliminate  t  from  any  two  of  these  equations  there  will  re- 
sult a  linear  equation  either  between  x  and  v,  or  between  x  and  z, 
or  else  between  y  and  z  ;  it  follows,  therefore,  that  the  centre  of  gra- 
vity of  the  system  must  describe  a  straight  line,  and  its  velocity  at 
any  time  will  be 


-J 


*"+*'+''^-  =  ./16»+6-.+6-^ 


df" 

which,  being  constant,  shows  that  the  motion  of  the  centre  of  gra- 
vity of  a  system  of  bodies,  whose  motions  are  entirely  due  to  their 
mutual  influences,  is  both  rectilinear  and  uniform. 

This  is  called  the  general  principle  of  the  conservation  of  the 
centre  of  gravity.     If  no  primitive  impulse  be  given  to  the  centre 
T  28 


2J8  ELEMENTS  OF  DYNAMICS. 

of  gravity  of  a  system,  then  b,  b',  b",  will  be  each  =  0,  and  there- 
fore, v=0,  or  the  centre  will  be  at  rest. 

(108.)  We  have  seen  at  (122,)  that  the  differential  expression 
ydx — xdy,*  is  the  difTerential  of  double  the  area  described  by  the 
projection  on  the  plane  of  xy  of  the  radius  vector  of  m  ;  hence  the 
sum  of  the  products  of  each  body  into  the  differential  expression  for 
the  area  it  traces  out  on  this  plane  in  any  time  will  be  expressed  by 
5  2  (mydx  —  mxdy).  In  like  manner  the  corresponding  sums  for 
the  other  planes  will  be  5  S  {mxdz — mzdx),  and  52  {mzdy  — 
mydz) ;  or  calling  these  several  sums  5  dS,  k  <iS',  5  dS",  and  diffe- 
rentiating, we  have 

2  {myd  'x  —  mxd  "y ) = rf  "S 

2  {mxd  ^z  —  mz  </  «x) = rf  2  S ' 

•zlmzd'^y  —  7)iyd^z)=d^S" 

rf'S 
and,  therefore,  from  the  equations  (B)  2  (myX  —  inxY)=-T—- 

2  {mxZ  —  mzX)  =  —^—  ;  2  (mzY  —  myZ)=  -j-  . 

Now  when  the  system  is  influenced  only  by  the  mutual  actions  of 
its  parts,  we  have  seen  (equa.  E,)  that  the  first  members  of  these 
equations  are  each=0,  consequently, 

*^  =»  •  -rf^  ="  •  l«r  ■••  S=<^' .  S=e'< ,  S"=c"( . 

This  result  establishes  the  general  principle  of  the  conservation  of 
areas,  viz.  that  in  any  system  of  bodies,  moving  in  virtue  of  their 
mutual  actions  on  each  other,  the  sum  of  the  products  of  each  body 
into  the  projection,  on  any  plane,  of  the  areas,  described  by  its  ra- 
dius vector,  is  proportional  to  the  time  in  which  those  areas  are 
described. 

(169.)  We  shall  investigate  one  more  general  principle,  which,  not 
being  deducible  from  the  general  equations  (A)  and  (B),  requires 
that  we  consider  the  motion  of  the  system  in  another  point  of  view. 

In  every  system  of  bodies,  the  motion  of  each  is  due  to  the  force 
impressed  on  it,  combined  with  those  which  arise  from  the  action 
of  the  other  parts  of  the  system  ;  the  simultaneous  action  of  all  these 
forces  determine  the  motion  of  any  one  body  m;  and,  therefore,  taking 
the  components  of  these  forces,  viz.  7nX,  mY,  mZ,  where  X,  Y,  Z 
are  the  accelerations  of  the  body  in  the  directions  of  three  rectangu- 
lar axes,  we  must  have,  at  any  time  t",  the  equations. 

*  The  sectorial  area  to  which  this  expression  appHes  is  measured  from  the  axis 
of  y,  as  at  art.  122  ;  but  for  the  complement  of  this  area,  or  that  measured  from 
the  axis  of  x,  the  corresponding  expression  is,  of  course,  xdt/  —  tfdx.  It  is  easy 
to  see  that  the  law  of  &reas,  established  in  this  article,  holds  in  either  case. 


MOTION    OF    A    SYSTEM    OF    BODIES.  219 

m  — ,—  =  m  X,  m  —,i^  =m  Y ,  m -5—  =m  Z 
at^  at-'  cir 

&c.  &c.  &c. 

Multiplying  now,  the  equation  in  x  by 

-T—  ,  the  equation  in  y  by  2  -y,  the  equation  in  z  by  2  —  ; 

dx 
and,  in  like  manner,  the  equation  in  x^,  by  2  -^,  and  so  on,  and  then 

adding  them  together  and  integrating,  we  shall  evidently  have  the 
equation 

dx'+dy'+dz'    ,        dx\  +  dy\-\-dz\    ,   . 

2fm  (X  dx+Y  dy+Z  rfz)+2/m,(X,  fZ:r,+Y,  c/y,+Z,  (/zj+&c. 

or,  in  the  usual  notation, 

dx^-\-dv'^-\-d'-^ 
2  (m       ^^^/        )=2  2/m  (XfZa;+Y  f/«/  +  Z  rfz) 

that  is,  s  (mi;'^)=2  ^fm(X  dx  +  Yf/y+Z  rfz)  .  .  .  .  (F),  which 
is  similar  to  the  equation  (2)  at  page  146,  when  the  system  consists 
of  but  a  single  point ;  the  second  member  is  also  integrable  in  like 
circumstances,  that  is,  when  X,  Y,  Z,  &c.,  are  functions  of  x,  y,  z, 
&c.  and  fulfil  also  the  conditions  named  at  page  146. 

If  these  conditions  have  place,  then,  as  at  the  page  referred  to, 
S(mD^)  —  •B(mv\)=f(x,y,  z,  x^,y^,z^.  Sic.)  — 
f  {a,  b,  c,  a^,  b^,  c^,  &c.) 
the  product  mv'^  of  the  mass  into  the  square  of  its  velocity  is  the  vis 
viva,  or  living  force,  so  that  when  the  second  member  of  (F)  is  an 
exact  differential,  we  infer  that  the  sum  of  the  living  forces  of  the  sys- 
tem generated  in  moving  from  one  position  to  another,  depends 
solely  upon  the  position  left  and  that  arrived  at,  and  is  independent 
of  the  paths  which  the  several  bodies  have  taken,  and  of  the  time  of 
describing  them.     This  is  the  general  principle  of  the  conservation 
of  living  forces. 

This  principle  always  holds,  that  is  to  say,  the  second  member 
of  (F)  is  always  an  exact  differential  when  the  bodies  move  in  vir- 
tue of  their  mutual  attractions.  To  prove  this,  let  r  be  the  distance 
between  two  of  the  bodies  of  the  system  m,  m^ ;  then 

y-={^x-xj^{y-yj+{z-z,y  .  .  .  .  (1)  ; 
let  Rbe  the  function  of  r  which  expresses  the  intensity  of  the  attrac- 
tive force  exercised  by  the  unit  of  m  on  m^,  and  by  the  unit  of  m^ 
on  m  ;  then  the  whole  attractive  force  of  m  on  m^  will  be  Rm  ;  and 
the  whole  attractive  force  of  m^  on  m  will  be  Rm^ .  The  compo- 
nents of  the  forces  Rm  will  be  (see  page  147,) 


220  ELEMENTS    OF    DYNAMICS. 


R,„^ ^  ,  Rm^ ^,  Rm- 


r        '  r       '  r 

and  these,  in  virtue  of  (1)  above,  are  the  same  as 

(ir  dr  dr 

Km  -f-  ,  Km  -7-  ,  Km  -r-  . 
dx  ay  dz 

In  like  manner,  for  the  components  of  the  force  Rm^ ,  we  have 

„        rf?*     „       </r  dr 

Rm,-^-.Rm.^,Rm,-^. 

This  last  set  of  components  liave  the  same  signs  as  the  former  set,  be- 
cause the  force  m,  R  is  opposite  to  7n  R,  and  the  coefficients  -; — ,  — — , 

dr       ,    .      .    .       .  .       .  ,  „  .         dr    dr    dr 

- —  ,  obviously  mvolve  opposite  signs  to  the  coemcients  -y-,  — ,  — . 

Substituting  the  foregoing  component  forces  for  X,  Y,  Z,  in  the 

expression  S  {in  X  dx-\-m  Y  dy-\-m  Z  dz),  we  have 

^dr    ,  ^-  dr    ,  _  rfr    ,  dr 

mmi  R-T-  (te+mmj  R-7-  dij-\-mm^  R -^  dz+m^mKj-^  tu", 

dr  dr 

-|-mimR-^ — (/(/i-f  m^  7?j  R  -^(/ci=mmi  Rrfr, 

the  first  side  of  this  equation  being  mm^  limes  the  total  ditTerential    fj 
of  J",  considered  as  a  function  of  the  six  variables  in  equation  (1) 
above.     The  same  reasoning  applied  to  every  two  bodies  in  the 
system,  must  lead  to  a  similar  result,  so  that, 

2  S  /  (m  X  dx+  mYdy  fm  Z  dz)=-i  1  .  ?n  f" ,  /  R  dr  , 
which  is  always  integrable,  since,  by  hypothesis,  R  is  a  function 
of  r. 


CHAPTER  VI. 

ON  THE  COMPOSITION  OF  ROTATORY  MOTIONS  ;  AND  ON  THE  PRINCIPAL 
AXES  OF  ROTATION  OF  A  SOLID  BODY. 

On  the  Composition  of  Rotatory  Motions. 

(170.)  If  a  body  receive  simultaneously  impulses  which  are  sepa- 
rately competent  to  produce  rotation  about  different  known  axes, 
the  result  will  be  a  rotation  about  a  new  and  determinable  axis.  It 
will  be  sufficient  to  consider  three  given  axes  at  right  angles  to  each 
other,  and  the  problem  we  propose  now  to  solve  is  this,  viz.  when 
the  body  tends  to  turn  at  the  same  instant  about  three  rectangular 
axes,  AX,  AY,  AZ,  with  the  respective  angular  velocities  «, «',  w"  ; 
to  determine  the  position  of  the  axis  about  which  it  actually  turns, 
and  the  angular  velocity  of  the  rotation. 


PRINCIPAL   AXES   OF    ROTATION.  221 

Let  m  represent  any  particle  of  the  body  (fiff.  125,)  and  let  us 
first  consider  the  motion  of  it  about  the  axis  AY  ;  this  will  be  in  a 
circle  pq7n,  and  we  shall  suppose  it  in  the  direction  of  these  letters. 
The  plane  of  the  circle  pqm  being  parallel  to  the  plane  of  xz,  the 
particle  has  no  motion  in  the  direction  of  AY,  and  its  motion  in  the 
directions  of  AX,  AZ  will  obviously  be  the  same  as  if  the  circle 
coincided  with  the  plane  of  xz,  we  may  then,  in  estimating  these 
motions,  suppose  such  to  be  the  case.  Now  as  m  turns  towards  the 
plane  of  xy,  its  co-ordinate  x  increases,  and  its  co-ordinate  z  di- 
minishes ;  hence  the  differential  of  x,  with  respect  to  the  time,  will 
be  positive,  that  of  z  negative.  The  absolute  velocity  of  in  about 
AY  will  at  any  time  t",  corresponding  to  the  co-ordinates  x,  y,  b(^ 
in  the  direction  of  the  tangent  mr,  taking,  therefore,  this  line  to 
represent  it,  its  components  in  the  directions  of  AX,  xlY,  will  be 
ms,  and  ma.  As  the  expression  for  the  absolute  velocity  is  Am  .  a>' 
we  have  Am  .  u'=^mr,  and  since  ms=^mr  sin.  inrs=mr  sin.  A= 
to'.  A?»sin.  A=u,'.mh=ia'  z,  this,  therefore,  expresses  the  velocity  in 
the  direction  of  AX  which  is  due  to  the  rotation  about  AY.  In  like 
manner,  ma=mr  cos.  rma=m7-  cos.  A^u' .  Am  cos.  A=u>'x  is  the 
velocity  in  the  direction  AZ  to  be  taken  negatively ;  hence  the  ve- 
locities in  the  directions  of  AX,  AZ,  due  to  the  rotation  of  any  par- 
ticle at  the  point  (x,  y,  z),  at  any  time  t"  is  w'  z  and  — ut'x. 

Consider  now  the  rotation  of  the  particle  m  about  the  axis  AX 
(fig.  126,)  from  Y  towards  Z,  and  applying  precisely  the  same  rea- 
soning, we  get  for  the  velocities  in  the  directions  of  Z  and  Y  the 
values  toy  and — i^z.  And  finally,  applying  the  same  reasoning 
when  the  rotation  is  about  the  axis  of  z  from  X  towards  Y  (fig. 
127),  we  have,  for  the  velocities  in  the  directions  of  AY,  AX,  w"  a: 
and  — "i"2/- 

It  follows,  therefore,  that  when  all  these  rotations  have  place 
simultaneously,  we  have,  by  adding  together  the  above  partial  velo- 
cities along  the  axes,  the  following  expressions  for  the  whole  velo- 
cities in  those  directions,  viz. 

dx       , 


-^=.z-.    y 

-^=w     X  —  wZ 

at 
dz 


(1)- 


Now  we  may  determine  from  these  expressions  about  what  axis 
the  body  actually  turns  at  the  instant  t",  when  the  foregoing  mo- 
tions have  place  simultaneously ;  for  as  every  particle  in  the  axis 
of  instantaneous  rotation  is  motionless,  we  must  have  for  all  of  them 
t2 


222  ELEMENTS    OF    DYNAMICS. 

dx     ^   (hi     ^    dz     ^ 

,¥=«•  i=»'  .¥=»■ 

that  is,  u,'  z  —  u"?/=0,  ^^"  X  —  wr=0,  wy  —  u'a:=0, 
and  these  three  equations  obviously  characterize  a  straight  line  in 
space  passing  through  tiie  origin  of  the  original  axes  ;  two  of  these 
equations  are  sufficient  to  determine  the  position  of  this  line,  as  for 
instance  the  equations 

a:=—  z,  y=—r,  z  .  .  .  .  (1),  where  -;-,  and  — ^ 

are  the  trigonometrical  tangents  of  the  angles  which  the  projections 
of  the  instantaneous  axis  make  with  the  axis  of  z ;  hence  if  a,  /3,  y, 
denote  the  angles  which  the  instantaneous  axis  of  rotation  makes 
with  the  axis  of  x,  y,  and  z,  we  have  {Anal.  Geom.  p.  228,) 


COS.  a=    .(    ^  ,     ,«  ,     ,771^  cos-  /3= 


cos.  y  = ; -, 

and  thus  the  position  of  the  required  axis  is  determined  in  terms  of 
the  known  angular  velocities. 

To  determine  the  angular  velocity  of  the  body  about  this  axis,  we 
need  consider  only  the  angular  velocity  of  any  single  particle  chosen 
at  pleasure ;  let  us  take  a  particle  on  the  axis  of  x ;  if  from  it  Ave 
draw  a  perpendicular  p  to  the  instantaneous  axis,  then  the  distance 
of  the  particle  from  the  origin  being  x,  we  have 

p=x  s\n.a=x<y\l — cos.*^    o= — ^--r -,. 

Now  as  for  this  particle  2/=0,  2=0,  in  the  second  members  of  (1), 
we  have,  for  its  absolute  velocity, 


v=V^ 


^x^-\-d\l^-\-dz^^  ,    ,„ 


and,  consequently,  for  the  angular  velocity  v,  we  have 

v=-=y/\^^^J^^J'^\ (2), 

p 
so  that  the  three  angidar  velocities  w,  to',  u",  about  three  rectangu- 
lar axes,  are  equivalent  to  the  singular  angular  velocity, 

about  an  axis  inclined  to  these  three,  at  angles  whose  cosines  are 
the  expressions  for  cos.  a,  cos.  3,  cos.  y,  above. 

It  is  obvious,  from  what  has  now  been  said,  that  when  a  body 
revolves  about  any  axis  given  in  position,  and  with  a  given  angular 
velocity,  we  may  always  resolve  the  motion  into  three  partial 
rotatory  motions  about  the  three  rectangular  axes  of  co-ordinates, 
these  component  motions  being  in  the  directions  assumed  at  the 


PRINCIPAL    AXES    OF    ROTATION.  223 

outset  of  this  article.  For  the  equations  of  the  axis  of  rotation  com- 
pared with  the  equations  (1),  and  the  expression  for  the  angular 
velocity  compared  with  the  expression  (2),  will  furnish  three  equa- 
tions among  the  unknowns  w,  w',  u",  which  are  sufficient  to  fix  their 
values.  Hence,  to  whatever  combination  of  rotatory  motion  the 
rotation  which  actually  has  place  be  due,  we  may  always  con- 
sider it  as  the  resultant  of  the  three  angular  motions  above  con- 
sidered. 

On  the  principal  Axes  of  Rotation. 

(171.)  There  are  some  remarkable  properties  belonging  to  cer- 
tain axes  of  rotation,  passing  through  any  point  in  space,  which 
well  deserve  notice;  for  instance,  whatever  point  be  chosen,  there 
always  exists  one  axis  in  reference  to  which  the  moment  of  inertia 
of  the  revolving  body  is  a  maximum,  and  another  for  which  the 
moment  of  inertia  is  a  minimum ;  these  two  axes  are  always  per- 
pendicular to  each  other,  and  they,  together  with  a  third  axis  througli 
the  same  point,  perpendicular  to  them  both,  are  called  the  three  prin- 
cipal axes  of  rotation,  passing  through  that  point.  When  the  point 
chosen  is  the  centre  of  gravity  of  the  body,  these  axes  have  each 
of  them  a  property  peculiar  to  themselves,  which  is  that  neither  of 
them  will  suffer  any  pressure  from  the  rotation  of  the  body  round 
it,  so  that,  when  this  rotation  has  once  commenced,  if  the  axis  were 
to  be  withdrawn,  the  rotation  would,  nevertheless,  continue  as  if  it 
were  there. 

To  establish  these  interesting  properties  it  will  be  requisite,  first, 
to  obtain  a  general  expression  for  the  momentum  of  inertia  of  a  re- 
volving body,  in  reference  to  any  axis  whatever. 

In  order  to  this,  let  AB  (fig.  128,)  be  the  axis  in  reference  to  which 
the  moment  is  to  be  determined,  and  assume  a  point  on  it  A  for  the 
origin  of  the  co-ordinates.  Let  /n  be  a  particle  of  the  body,  and  of 
which  the  co-ordinates  are  x,  y,  z ;  let  the  perpendicular  niB  =  r, 
and  the  distance  A.in  of  the  particle  from  the  origin  6 ;  also  call  the 
angle  mkB,  e  and  let  a,  /3,  y ;  a',  i3',  y',  be  the  angles  which  AB, 
and  Am,  make  with  the  axes  of  x,y,  z;  we  shall  thus  have 

82  =  a;3  4-y-|-23.   ...    (1) 

cos.  e  =  cos.  a  cos.  a'  +  cos.  j3  COS.  j3'-|-cos.  y  cos.  y' ; 

therefore,  since  cos.  a'  =  — ,  cos.  3'  =— ,  cos.  y'  =— ,  we  have, 

by  substitution,  5  cos. «  =  x  cos.  a-\-y  cos.  |3-f  z  cos.  y  .  (2). 

Now  in  the  right-angled  triangle  mAB  we  have  r^  =  5^  sin.  ^  s=  S'' 
— 5^  COS.  ^  s ;  therefore,  substituting  in  this  value  of  r^  the  expres- 
sions (1)  and  (2),  we  have  r^  =  (1  —  cos.^  a)  x^-{-(l-{-cos.  ^^)  y"* 
-f-   (1 — cos.  '^y)  z^ — 2xy  COS.  a  COS.  (3 — 2  xz   cos.   »   cos.  y — 


224  ELEMENTS  OF  DYNAMICS. 

2  yz  COS.  /3  COS.  y,  that  is,  r'=:x'  sin.  '  a+y"  sin. '  /3-f-2'  sin.  "  ■y — 
2  ary  cos.  a  cos.  /} — 2  xz  cos.  o  cos.  •y — 2  yc  cos.  ,3  cos.  y. 

From  this  equation  we  immediately  get  the  expression  for  the 
moment  of  inertia  2  (r^  m)  of  the  body,  in  reference  to  the  axis 
AB,  for  putting  for  brevity 

2  {x^m)=k,  2  (v2  w)=B,  2  (z2m)  =  C 
2  (art/m)=D,  2  (a'2-m)=E,  2  (yzw)=F 
the  foregoing  equation  gives 

2  {r^m)=k  sin.2a  +  B  sin.*  ;3  +  C  sin.^y  J         _ 

— 2D  cos.acos./3 — 2  E  cos.  a  cos.  y — 2Fcos.|3cos.y5  ^  ' 
which  is  the  general  expression  for  the  moment  of  inertia  of  the 
mass  M,  in  reference  to  any  axis  AB  through  the  origin  of  the  co- 
ordinates, and  inclined  to  them  at  the  angles  a,  /3,  y.  The  position 
of  the  rectangular  axes  of  co-ordinates  originating  at  A,  being  ar- 
bitrary, if  we  could  so  fix  them  that  they  should  give  D=0,  E=0, 
F=0  ....  (4),  the  general  expression  (3)  would  be  considerably 
simplified,  being  for  such  axes  reduced  to  the  first  line ;  that  is,  we 
ijhould  then  have, 

2  (r27/i)=Asin.2a-fBsin.2^-fCsin.2y  ....  (5). 
We  shall  presently  show  that  there  really  does  exist  for  every 
point  A  three  rectangular  axes,  for  which  the  conditions  (4)  have 
place,  these  being  what  are  called  the  principal  axes  of  rotation. 
But,  before  we  enter  upon  the  proof  of  this,  it  will  be  expedient  to 
show  how  to  transform  the  equation  (3)  into  another,  into  which 
there  shall  enter,  instead  of  A,  B,  C,  the  expressions  for  the  mo- 
ments of  inertia  around  the  three  axes  chosen  for  those  of  the  co- 
ordinates. Now  as  the  distance  r'  of  the  particle  m  from  the  axis 
of  .r  is  v'  \y^-\-~^  |»  its  distance  r"  from  the  axis  of  y,  v/la^+^*|» 
and  its  distance  r'"  from  the  axis  of  r,  ^\x'^-\-y''\,  it  follows  that 
if  we  put  for  the  moments  of  inertia  about  these  axes  the  symbols 
A',  B',  C',  we  shall  have 

2  (r'^  »i)=2  (y^-\-z^)  m=B  +  C=A' 

2  (r"2m)=2  (x'^+z^)  7n=A+C=B' 

2(r""'m)=2  (ar^-fy'')  w=A+B=C'. 
When,  therefore,  these  three  moments  of  inertia  are  known,  the 
moment,  with  respect  to  any  other  axis  AB,  will  be  obtained  by 
substituting  in  the  equation  (3)  the  values  of  A,  B,  C,  in  terms  of 
A',  B',  C',  deduced  from  these  expressions;  the  result  of  this  sub- 
stitution is 

2  {r^m)=h  A'  (sin.  ''/S-fsin.^y  —  sin.'^  a) 
-f  i  B'  (sin.=  o-fsin.=  y  — sin.  ^  ^)4-4  C  (sin.*  o+sin.«^  —  sin.^  y). 
But  since  cos.  *  a  -|-cos.  *  ,3-|-cos.  *  y=l  ....  (6),  or  1  —  sin.  *  o 
-f-1  — sin.*  i3-\-l  —  sin.*  y  =1,  or  sin.-  a+  sin.*  /3  -f  sin.*  y=2; 
this  equation  is  the  same  as  2  (r*m)=A'  cos.  *  a  +  B'  cos.  '  /3-f- 
C'cos.^^ (7)  ;  which  shows  that  the  moment  of  inertia,  with 


7?7 


Jl'l  _       /'/„/,  II 


-TFl 


in 


PRINCIPAL   AXES   OF    ROTATION.  225 

respect  to  any  axis,  is  equal  to  the  sum  of  the  products,  arising  from 
multiplying  each  of  the  moments,  with  respect  to  the  principal  axes 
by  the  squares  of  the  cosines  of  the  inclinations  of  the  proposed  axis 
to  these.  On  account  of  the  relation  (6)  between  these  inclinations, 
we  need  introduce  but  two  of  them  into  the  expression  (7),  which 
may  be  written 
S(r2m)=A'+(B'— A')cos.2  ^+(C' —A')  cos.^  y  .  .  .  .  (8). 
The  quantities  A',  B',  C',  are  necessarily  positive,  being  formed 
from  squares  multiplied  by  masses :  hence,  if  A'  is  the  smallest  of 
the  three,  every  term  in  this  equation  will  be  positive,  and,  what- 
ever the  arbitrary  angles  j3,  y  may  be,  we  must  always  have,  in  this 
case,  2  (r^ni)  >  A',  that  is  to  say,  no  other  line  AB  through  the 
origin  can  be  found,  about  which,  if  the  body  revolve,  the  moment 
of  inertia  can  be  so  small  as  when  the  body  revolves  about  that 
principal  axis  which  we  have  taken  for  the  axis  of  x.  But,  if  A'  is 
the  greatest  of  the  three  quantities,  then  S  (r^m)  <;A',  for  every 
value  of  /3  and  y,  so  that  then  the  principal  axis  in  question  will  be 
that  for  which  the  moment  of  inertia  is  greater  than  for  any  other 
axis  through  the  same  origin.  A'  may,  however,  be  neither  the 
greatest  nor  the  least  of  the  three  moments  A',  B',  C,  but  may  be 
intermediate  between  B',  C';  but  we  may  make  either  of  these 
three  quantities  stand  first  in  the  equation  (8),  by  eliminating 
from  (7)  that  angle  which  multiplies  the  quantity  we  wish  to  stand 
alone,  by  means  of  the  relation  (6),  thus  if  y  be  eliminated  instead  of 
a,  as  above,  then  C'  will  stand  first,  and  the  same  conclusions  will 
then  apply  to  the  axis  of  z  instead  of  to  the  axis  of  x  ;  hence,  of 
the  three  moments  of  inertia  relative  to  the  principal  axes,  one  of 
them  is  a  maximum,  and  another  a  m,inimi(m. 

This  conclusion,  however,  is  on  the  supposition  that  the  princi- 
pal moments  are  all  unequal ;  but  it  may  happen  that  this  is  not  the 
case ;  let  us  suppose  then  that  two  of  them  are  equal,  as  A'=B', 
then  the  equation  (8)  becomes 

S  (r2m)=A'  +  (C'  —  A')  cos.^'y  ; 
where  it  is  evident  that  if  A'>  C,  A'  will  always  be  >  S  (r^m), 
provided  y  is  not  90°,  and-,  with  the  same  condition,  A'  will  always 
be  <  2  (r'^m),  if  A'  <CC' ;  but,  when  y=90,  whatever  a  and  /3  be, 
then  always  2  (r^m)=A';  we  infer,  therefore,  that  when  the  prin- 
cipal moments  of  inertia  relative  to  the  axis  of  x  and  y  are  equal, 
the  moments  are  all  equal,  for  every  axis  in  the  plane  of  xy,  and 
the  moment  relative  to  the  axis  of  z,  will  be  a  maximum  or  a 
minimum,  according  as  A'<C',  or  A'>C'. 

If  A'=B'  =  C',  then  2  (r2m)=A',  so  that,  in  this  case,  all  the 
axes  through  the  origin  are  principal  axes.  It  follows,  therefore, 
that  when  more  than  three  principal  axes  can  oass  through  any 
point,  an  infinite  can  pass  through  that  point, 

29 


226  ELEMENTS  OF  DYNAMICS. 

(172.)  It  is  time  now  to  show  tliat  tliroiigli  every  point  of  space 
three  principal  axes  may  always  he  tlrawii ;  that  is,  three  axes  in 
reference  to  which  the  equations  (4)  have  place.  From  the  pro- 
perties just  developed  we  shall,  obviously,  he  led  to  one  or  other 
of  these  principal  axes,  if  they  exist  by  determining  for  what  values 
of  the  arbitrary  and  independent  angles  o,  p,  the  general  expression 
(3),  for  the  moment  relative  to  any  axis,  becomes  a  maximum  or  a 
minimum  ;  so  that,  for  the  determination  of  the  position  of  the  axes 
of  greatest  or  least  moment  or  of  the  suitable  values  of  a  and  li,  we 
should  have,  by  the  theory  of  maxima  and  minima,  the  two  equa- 

dx(r'm)     „   dx(r'm)      „       ...  ^  ■     .      ^      , 

lions -. ^=0,  ^i ^=0;  which  are  sufficient  to  fix  the 

da  0/3 

values  of  a  and  ,3.  The  performance  of  the  actual  differentiation, 
here  indicated,  is  an  easy  matter ;  but  the  subsequent  elimination 
of  a  or  of  j3,  in  order  to  obtain  a  final  equation  involving  only  one 
of  these  quantities,  is  a  very  tedious  and  troublesome  operation, 
which,  at  length,  conducts  to  a  complicated  cubic  equation.  In- 
stead, therefore,  of  employing  this  method  of  investigation,  we 
shall,  after  the  example  of  Professor  Whcwell,  adopt  the  following 
shorter  and  more  elegant  process  from  Lagrange.  The  problem 
is  to  find  the  position  of  three  axes  of  rectangular  co-ordinates, 
x',  y',  z',  such  that 

2  (x'  y'  m)=0,  2  {x'  z'  m)=0,  2  {y'  z'  m)=0  ....  (A). 
Let  the  three  fixed  axes,  in  reference  to  which  the  required  ones 
are  to  be  determined,  be  those  of  x,  y,  z,  both  systems  having  the 
same  origin. 
Let  x'  make  with  x,  y,  z,  angles  whose  cosines  are  a,   b,    c   1 

y'  .  .    •         .  .a',  b',  c'   [-.(1); 

z'         .  .  .  .  a",6",c"J 

then  the  angles  contained  between  x  and  y,  between  x  and  ?,  and 
between  y  and  z,  being  right  angles,  and,  consequently,  their  co- 
8ine8=0,  we  have  (Anal.  Geom.  p.  228,  art.  182.) 
a  a'  +bb'  +cc'  =0~] 
aa"-\-b'b"+cc"=0 
a'a"  +b'b"-\-c'c"=0 
and  (Anal.  Geom.  p.  228,  art.  181,)  ^  ....  (2); 
a«  -f6'»  +0"  =1 
a'»+6'«  -j-c'^  =1 
a"3_j-6"3-|_c"«=l. 
and  these  are  the  six  equations  of  condition,  which  must  be  ful- 
filled by  the  constants  (1).     Now  we  have  already  seen  that  the 
general  expression  for  the  moment  of  inertia,  with  respect  to  any 
axis  AB,  making  the  angles  a,  )3,  y,  with  x,  y,  z  is 


PRINCIPAL    AXES    OF    ROTATION. 


227 


2(r''m)  =  Asin. ''a  +B  sin. '2)3  +  0  sin.  «  y. 

—  2D  COS.  a  COS.  /3  —  2E  COS.  a  COS.  y  —  2  F  cos.  j3  cos.  y. 
But  if  this  same  axis  AB  make  the  angles  a',  fi',  y',  with  x',  ii'  z' 
then  since,  by  hypothesis,  2  {x'y'ni)  =0,  &c.  the  moment  of  in- 
ertia, in  reference  to  it  will  be 

2  {r^m)=k  sin.  2a'  +  B'sin.  2;3'-|-C'sin.  ^y'; 
where  A',  B',  C,  stand  for  S  {x'^m),  2  {y'^m,)  2  (z"^m.) 

These  two  expressions  for  S  (rVi)  are,  therefore  equal.     The 
latter  is  the  same  as 

A'  +  B'  +  C  — A'cos.  2  a'— B'cos.2|3'  — C'cos.2  y', 
but  A'  +  B'  +  C'  =  S  (  x'^+y'-^^z"')  m=s  {x"~+y''+z^)  m ;  and  this 
last  expression  is  what  we  have  before  represented  by  A  +  B  +  C  ; 
hence,  substituting  in  the  first  expression,  A  —  A  cos.^a  for 
Asm.i^tt,  B  —  B  cos. 2  a  for  B  sin.  ^  /3,  and  C  —  cos.  ^  y  for  C  sin.  ^  y, 
and  then  equating  the  two,  we  have 

A  +  B  +  C  — (A  COS. 2  a+B  cos.2|3  +C  cos."  y) 
—  (2D  COS.  a  cos.  ,3+2  E  cos.  a  cos.  y+2  F  cos,  p  cos.  y) 
=A  +  B  +  C  — (A'cos.  2a'  +  B'cos.2/3'  +  C'  cos.^  y')  ....  (3.) 
Now  since  (Anal.  Geom.  p.  228,  art.  182,) 

COS.  a'^a  cos.  a+6   cos.  )3  +  c    cos.  y 
cos.  ^'■:=a'  cos.  a+6'   COS.  ^-\-c'   cos.  y 
COS.  y'=a"  cos.  a+6"  cos.  j3  +  c''  cos.  y  ; 
the  last  term  in  (3)  becomes 

A'  (a^  COS.  ^  a-\-b^  cos.  2  p  +  c'^  cos.  ^  y 

+  2  «&  cos.  tt  COS.  3+2  ac  cos.  a  cos.  y+26c  cos.  /3  cos.  y) 

+  B'  (a'2  COS.  2  a+6'2  cos.  «  |3  +  c'  2  cos.  "  y 

+2  ft'  6'  cos.  a  cos.  j3+2  a'  c'  cos.  a  cos.  y+2  b'  c'  cos,  j3  cos.  y) 

+  C'  {a"^cos.  a+6"=*cos.2/3+c"2cos.2y 

+2  a"  6"  cos.  a  cos.  ,3+2  a"  c"  cos.  acos.  y+2  b"  c"  cos.  /3cos.y); 

and  this  expression  must  be  identical  with 

A  COS.  "a+B  COS.  ^/3  +  B  cos,  "y 

+  2  D  COS.  a  COS.  i3  +  2  E  cos.  a  cos.    y  +  2  F  COS.  /3  COS.  y ; 

hence,  equating  the  co-efficients  of  the  like  terms,  we  have 

A'a2+B'a'^+C'a"2    =A (1') 

A'b^ +B' b'^ +C'b"^    =B (2') 

A'c^+B'c'^+C'c"^    =C (3') 

A'ab+B'a'b'-i-C'a"b"=T) (4') 

A'oc+B'a'c'  +  C'a"c"==E  ....  (5') 
A'6c-fB'6'c'+C'6"c"=F.  ,  .  .  (6') 
These  six  equations,  combined  with  the  six  marked  (2),  are 
sufficient  to  determine  the  twelve  unknown  quantities  which  enter 
them,  but  we  shall  only  require  to  determine  four  of  them,  viz. 
n,  b,  c,  and  A',  and  shall  therefore  eliminate  the  rest.  In  order  to 
this,  add  together  (1')  a,  (4')  b,  (5')  c,  and  we  have 


228  ELEMENTS  OF   DYNAMICS. 

A'a(a«+i«-f  c«)+B'a'  (aa'+bb'-\-cr')  +  C  a"  (aa"+bb"  +  cc")  = 

Aa+Dft+Ec; 
or,  by  the  conditions  (2),  page  226, 

A'fl=Ao+D6-fEc') 
Similarly  (2')  6  +  (6')  c+(4')  a  gives  A'6=BZ;  +  Fc-t-Dfl  [  .  .  (4). 

.     .      (3')  c  +  (5')  a  +  (6')  6  A'c=Cc  +  Ea  +  F6  J 

These  three  equations,  together  with  the  condition  a'-f  6'+c*= 
1,  are  sufficient  to  determine  a,  b,  c,  and  A'. 
By  the  first  two  (A' — A)  a  —  Db  —  Ec  =0 
(A'  — B)6— Fc  —  Da=0 
from  which,  by  eliminating  c,  we  have 

KA'  — A)F+ED^  «— |(A'  — B)E  +  FD|6=0 
■  ^,_(A'-A)F+ED 
••^-(A'_B)E  +  FD"';"^^^' 
also  eliminating  b  from  the  same  two  equations, 

KA'  — A)(A  — B)  — D-^  a— KA'  — B)E+FD|  c=0 
.  .      (A'-A)(A'-B)-D^ 

••'=        (A'-B)E+F1) ""•••'  ^^)- 

Substituting  these  values  of  b  and  c  in  the  third  of  equations  (4), 
that  is,  in  (A'  —  C)c  —  Ea  —  F  6=0,  there  results 

(A'-A)(A'-B)-D^  (A'-A)F+ED 

C^  -^^      (A'-B)E  +  FD         ^-*  (A^ITBTEirFD-"' 
or,  (A'  — A)(A'— B)(A'  — C)  — 

{(A'  —  A)  F^'+CA—  B)  E*-|-(A'  —  C)  D«|  —  2  FED=0 (7), 

a  cubic  equation  in  A'.  This  equation  being  of  the  third  degree 
has  necessarily,  at  least,  one  real  root,  and  consequently,  the  values 
of  a,  b,  and  c,  as  determined  from  the  equations  (5)  and  (6)  com- 
bined with  the  equation  a^-^b^-^c'^=l,  are  real,  so  that  there  exists 
at  least  one  principal  axis,  viz.  the  axis  of  x',  and  its  position  is 
determined  by  these  three  equations. 

Returning  now  to  the  equations  (1'),  (2'),  &c.  and  making  the 
same  combinations  of  them  as  before,  only  using  now  a',  b',  c', 
instead  of  a,  b,  c,  we  shall,  obviously,  be  led  to  the  same  cubic 
equation  (7),  except  that  B'  will  occupy  the  place  of  A'  ;  and  if  we 
use  a",  6",  c",  instead  of  a,  b,  c,  the  resulting  cubic  will  differ 
from  that  above  only  in  having  C  in  place  of  A'.  Thus  although 
the  first  of  these  cubics  determine  three  positions  for  the  axis  of  x' 
(one  at  least  being  real)  ;  the  second,  three  positions  for  the  axis 
of  y' ;  and  the  third,  three  positions  for  the  axis  of  z' ;  yet  these 
systems  of  threes  must  coincide,  and  can,  therefore,  furnish  only 
three  distinct  and  different  directions  for  the  axes  of  x',  y',  and  z', 
given  by  the  three  roots  of  (7). 

It  remains  to  show  that  these  roots  are  all  real.    Suppose  two  of 


PRINCIPAL   AXES    OF    ROTATION.  229 

them  to  be  impossible,  and,  therefore,  of  the  form  m  +  n'^'^l, 
and  m  —  n\/  —  1 ;  the  quantities  a,  b,  c,  are  possible,  when  the  root 
A'  is  so,  and  for  one  of  the  impossible  roots  the  corresponding  quan- 
tities a',  b',  c',  will  be  of  the  form  p+q  "^ —  1,  p'+q'  v^^H^, 
p"-]-q"->/  —  1,  and  for  the  other  root,  a",  b",  c",  will  be  of  the 
formju  —  q  >/  —  1,;j'  —  q'  ^^ —  \,p"  —  q"  ^  —  1.     Now  as 
a'  a"-j-b'  b"  -\-c'  c"=0 
.•.p''-i-p">+p"^+q^+q'^-\-q"^=0, 
which  is  absurd,  because  the  sum  of  a  series  of  squares  can  never 
be  0.     Hence  the  three  rectangular  axes,  each  having  the  property 
(A),  really  exist  through  whatever  point  we  require  them  to  be 
drawn,  because  the  origin  may  be  arbitrary. 

(173.)  But  the  existence  of  the  three  principal  axes  may  be  es- 
tablished in  another  way,  after  having  shown,  as  above,  that  one 
exists.  For  suppose  the  axis  of  x,  whose  position  is  arbitrary,  to 
coincide  with  this  principal  axis ;  then  we  must  have  o=l,  b=0, 
c^O,  and  these  values  substituted  in  the  three  equations  (4)  in  a,  b,  c, 
above,  give  A'' — A^O,  D=0,  E=0,  so  that  the  cubic  equation  (7), 
from  which  the  values  of  A'  are  to  be  deduced,  becomes,  by  putting 
these  values  for  D  and  E, 

(A'  —  A)  (A'  —  B)  (A'  _  C)  —  (A'  —  A)  F==0 

.-.  (A'  —  B)  (A  —  C)  —  F^=0, 
or  A'^  — (B  +  C)  A'  +  BC  -  F=^=0 (1). 

This  being  a  quadratic  will  furnish  two  values  for  A',  and  to  de- 
termine the  corresponding  inclinations  a,  b,  c,  we  have  (4)  the  two 
equations  A'  b^Bb-\-Fc-{-Da,a"-\-b^-{-c^=l ;  but  D=0,  and,  since 
the  axis  to  be  determined  must  make  a  right  angle  with  that  of  x, 
we  must  have  a=0,  therefore 

(A'  — B)6=Fc;  b^-i-c^  =  l, 
so  that  if  d  represent  the  inclination  of  the  required  axis  to  that  of 

,       ,  c     A'  — B 

y,  then  o=cos.  6,  c=sm.  9,  .*.  tan.  0=— = — — — 

_    2  tan.  0    _    2  F  (A'  —  B) 
•'•  t^"-  2  ^=TZ-[^Te~F^  _(A'  —  B^  ' 
the  denominator  of  this  fraction  will  be  given  by  adding  the  identi- 
cal equation  (C  _  B)  A'  —  (C  —  B)  B=(A'  —  B)  (C  —  B)  to  the 
equation  (1),  for  there  results 

(A'_B)2  — F^=(A'  — B)  (C  — B) (2). 

2  F 
.-.  tan.2  9=g--^ (3), 

and,  as  the  same  tangent  belongs  to  two  arcs  of  which  the  differ 
ence  is  180°,  therefore  there  are  two  values  for  9,  of  which  the  dif- 
ference is  90°,  so  that  besides  the  principal  axis,  which  has  been 
U 


230  ELEMENTS    OF    DYNAMICS. 

made  coincident  with  that  of  x,  there  are  two  otlicrs  in  tlio  plane  of 
zy,  inclined  to  the  axis  of  y  in  the  angles  e  and  90-f-e,  or  perpen- 
dicular to  each  other. 

AVhen  we  know  the  position  of  one  of  the  principal  axes,  taking 
it  for  the  axis  of  x,  the  position  of  the  other  two  becomes  deter- 
minable from  tlie  equation  (3),  just  deduced. 

(174.)  Let  us  now  prove,  that  if  a  body  revolve  about  one  of  its 
principal  axes  passing  through  the  centre  of  gravity,  this  axis  will 
suffer  no  pressure  from  the  centrifugal  forces  of  the  several  particles. 

Let  the  body  revolve  about  the  axis  of  z,  then  every  particle  m 
will  describe  about  this  axis  the  circumference  of  a  circle  of  radius 
s/\x'^-\-y- 1  and,  therefore,  if  u  be  the  angular  velocity  of  the  sys- 
tem, «  \/^x^+i/2  \  will  express  the  rotatory  velocity  of  any  particle 
7/1  whose  co-ordinates  are  x,y  \  but  the  centrifugal  force  being  equal 
to  the  square  of  tlie  velocity,  divided  by  the  radius,  its  general  ex- 
pression here  is  co^  \/\x'^-^y'^\  and  consequently  the  strain  which 
any  particle  m  produces  on  the  axis  is  m  (^^  y/ \x'^ -\-y- \  ;  if  this 
force  be  resolved  in  directions  parallel  to  x  and  y,  the  two  compo- 
nents will  be  mu>^ X  and  mu'^y,  and  the  moment  of  these  forces,  to 
turn  the  body  about  the  axis  of  ^  and  o(  x,  will  be  inu^'xz  and  rriw^yz, 
and  therefore,  of  the  forces  exercised  by  all  the  particles,  the  mo- 
ments will  be 

u'^  2  (w  xz)  and  o^^  2  (myz)  ....  (1), 
if  these  be  each  0,  there  will  be  no  effort  used  by  the  centrifugal 
forces  to  incline  the  axis  of  z  towards  the  plane  of  xy ;  such  is  the 
case,  therefore,  when  the  axis  of  rotation  is  a  principal  axis  ;  hence, 
in  this  case,  the  only  effect  of  the  forces  mu^x  and  mts'y  on  the  axis, 
is  to  move  it  parallel  to  itself,  or  to  translate  the  body  in  the  direc- 
tions of  X  and  y  ;  the  aggregate  of  these  forces  is 

w^  2  (m  x)  and  u«  2  (m  y)  .  .  .  .  (2), 
and  if  these  be  each  0,  the  forces  will  use  no  effort  to  move  the 
body  or  to  press  the  axis :  and  they  are  0  when  the  axis  passes 
through  the  centre  of  gravity ;  we  conclude,  therefore,  that  when  a 
body  revolves  about  one  of  its  principal  axes  passing  through  the 
centre  of  gravity,  the  rotation  causes  no  pressure  whatever  upon 
the  axis,  which  may,  therefore,  be  removed  without  at  all  affecting 
the  motion  of  the  body,  the  rotation  once  impressed  continuing  per- 
manent. On  this  account  the  principal  axes  through  the  centre  of 
gravity  are  called  the  axes  of  permanent  rotation,  or  by  some,  the 
natural  axes  of  rotation. 

(175.)  If  the  initial  rotatory  motion  of  the  body  be  not  about  a 
permanent  axis  of  rotation,  the  effects  of  the  centrifugal  forces  on 
the  axes  cannot  be  destroyed,  inasmuch  as  the  foregoing  conditions 
cannot  obtain ;  these  forces,  therefore,  will  alter  the  axis  of  rotation, 
and  the  body  will  at  every  instant  of  its  motion,  if  free,  turn  about 


PRINCIPAL   AXES   OF    ROTATION.  231 

a  different  axis,  called  the  instantaneous  axis  of  rotation  ;  and  it 
may  be  proved  that  if  this  axis  does  not  at  the  commencement  of 
motion  coincide  with  a  permanent  axis,  it  can  never  coincide  with 
one  afterwards,  so  that  whenever  we  observe  a  body  to  revolve 
about  one  axis  during  any  time,  however  short,  we  may  conclude 
that  it  has  continued  to  revolve  about  that  axis  from  the  commence- 
ment of  the  motion,  and  that  it  will  continue  to  revolve  about  it  for 
ever,  imless  checked  by  some  extraneous  obstacle. 

These  particulars  will  be  more  completely  established  in  the  fol- 
lowing articles. 

We  may  further  remark  here,  that  when  a  body  revolves  about 
any  one  of  the  principal  axes  passing  through  the  fixed  point,  which 
is  taken  for  the  origin,  although  this  point  be  not  the  centre  of  gra- 
vity of  the  body,  yet  the  expressions  (1)  will  still  be  0,  so  that  the 
revolving  body  will  use  no  effort  to  cause  the  axis  to  turn  about  the 
origin  in  any  direction ;  but  as  the  expressions  (2)  will  not  be  0,  the 
axis  must  sustain  a  pressure  in  the  directions  of  x  of  y,  which  would 
cause  a  tendency  in  the  axis  of  z  to  turn  about  those  of  y  and  x, 
unless  these  pressures  were  wholly  exerted  upon  the  fixed  point  or 
origin ;  that  is,  unless  the  resultant  of  all  the  pressures  passed  through 
this  point ;  such,  therefore,  in  virtue  of  the  former  conditions,  must 
be  the  case ;  so  that  through  any  given  fixed  point  in  a  body,  there 
may  always  be  drawn  three  axes  around  which  the  body  may  turn 
uniformly  without  changing  its  original  axis  of  rotation,  although  it 
would  be  at  liberty  to  do  so,  as  it  is  free  to  move  in  any  direction 
about  the  fixed  point.  In  order,  therefore,  that  a  body  retained  at 
rest  by  a  single  fixed  point  may,  by  means  of  an  impulse,  receive  a 
permanent  motion  of  rotation,  it  is  necessary  and  sufficient  that  the 
impulse  be  such  as  to  cause  a  percussion  on  one  of  the  principal 
axes  of  the  body,  through  the  point,  equivalent  to  a  single  force  ap- 
plied to  this  point  perpendicularly  to  the  same  axis. 

It  remains  now  for  us  to  prove  the  assertion  above,  viz.  that  the 
instantaneous  axis  of  rotation  can  at  no  instant  coincide  with  a  per- 
manent axis  unless  the  body  has  continued  to  revolve  about  this 
axis  from  the  commencement  of  the  motion,  and  in  order  to  this  it 
will  be  convenient,  first,  to  ascertain  the  equations  which  express 
the  general  theory  of  a  body's  rotation  about  its  centre  of  gravity, 
and  then  to  discuss  those  particular  forms  of  them  which  arise  from 
supposing  the  rotation  to  take  place  about  the  principal  axes. 

(176.)  The  group  of  equations  (A,  B,)  at  art.  (166),  which  ex- 
presses the  conditions  of  the  motions  of  any  system  of  bodies  mutu- 
ally connected,  and  each  acted  upon  by  any  accelerative  forces, 
obviously  holds  when  the  system  constitutes  a  solid  body  ;  we  may 
regard  them,  therefore,  as  embodying  the  analytical  theory  of  the 


232  ELEMENTS  OF  DTNAHICS. 

motion  of  a  solid  body,  of  which  each  particle  is  acted  upon  by  any 
accelerative  forces. 

The  first  three  of  the  equations  referred  to,  completely  determine 
the  progressive  motion  of  the  body  in  space,  or  the  path  described 
by  its  centre  of  gravity,  furnishing  for  this  purpose  the  requisite 
equations 

rf'x  _  2(?nX)   rf'Y  _  s(mY)    d^z  _  s  (rnZ) 

dt^  ~       AT"'  (//2  ~        M       '  "^  M~' 

where  the  sign  S  includes  under  it  all  (he  particles  m,  m,,  m,,  &c. 

of  the  mass  M,  which  are  acted  upon  by  the  accelerative  forces  X, 

X„X,;  Y,  Y,,  Y„«fec. 

The  remaining  three  equations  (B)  must  be  those  which  deter- 
mine the  rotatory  motion  of  the  body  round  this  moving  centre  ;  or 
if  the  centre  of  gravity  remain  fixed,  and  the  body  be  free  to  move 
round  it  in  every  direction,  then  the  three  equations  (B)  must  be 
sufficient  to  determine  the  circumstances  of  the  rotatory  motion 
arising  from  the  action  of  the  same  accelerative  forces. 

The  rotatory  motion  thus  produced,  on  the  supposition  that  the 
centre  of  gravity  is  fixed,  must,  since  the  progressive  and  rotatory 
motions  are  independent  of  each  other,  be  really  that  which  accom- 
panies the  progressive  motion  the  body  actually  has  when  this  cen- 
tre is  free,  and  which  progressive  motion  is  that  which  the  centre 
would  have  if  all  the  accelerative  forces  acted  upon  it  as  a  single 
free  point,  so  that  the  absolute  motion  in  space  of  ai  particle  of 
the  body  is  compounded  of  these  two. 

Supposing  then  the  centre  of  gravity  of  the  body  t  be  fixed,  if 
we  place  there  the  origin  of  the  co-ordinates,  all  that  .oncerns  the 
rotatory  motions  of  the  body  will  be  comprised  in  the  equations 

.  y  d"  X         X  d^  y     .         ,^  _.        . 

S  (-^^^^     m ^^w)=S  (Xym  —  Yx  m) 

^^~di^'^ 'W^^^^  (ZxTW  — Xzm) 

z  d^  y         11  d^  z 
S(-^m-^^^^^m)=S(Yz/n  — Zy7n). 

Let  us  represent,  as  at  (170)  the  angular  velocities  of  the  body 
about  the  three  axes  of  co-ordinates  by  w,  w',  w",  then,  instead  of 

the  coefficients -y-j ,  -r-j  ,  we  may  introduce  their  values  from  equa- 
tion (1),  page  221,  so  that  the  first  of  the  preceding  equations  will 
be,  by  transposing, 

_ /"ir             V        N           d(o"x  —  iiz)                 d(u'z  —  «"y) 
X(Ya?m — Xi/m)=2x— i ^ -m  —  sy — ^^ — -3- ^^m. 


PRINCIPAL    AXES    OF    ROTATION.  233 

If  we  actually  perform  the  differentiation  here  indicated,  and  always 

,     .         .     dx  dy   dz    .    . 
substitute  for  — ,  ^,  — ,  their  values  as  given  by  the  equations  re- 
ferred to,  we  shall  have 

(2co"2+co'2— co2)  2  (xym) 
— (»""+-^)  S  (y^  m)  — («"«'+  £)^{xz  m)+ 

o  «'  S  (x^—y^)  m. 
Now  let  us  suppose  the  axes  of  co-ordinates  to  be  the  principal 
axes  of  the  body,  then  we  know  that 

2  (xym)=0,  (xzm)=0,  {yzm)=0; 
hence,  putting 

2(y2+z2)wi=:A,  2G'r2+z2)m=B,  ^(x^ -\-y'')  m=C 
.-.  2(a?2-^2)  7n=B  — A, 
the  foregoing  equation  becomes  simply 

2:(Ya?m  — X2/m)  =  C-^+(B  — A)  ««', 

and,  in  a  precisely  similar  way,  we  obtain 

2  (X  z  m— Z  a;  wi)=B-^ -f  (A  —  C)«w" 

2(Zym  — Yzw)=A^+(C  — B)«'«". 

(177.)  Suppose  now  that  no  accelerative  forces,  X,  Y,  &c.  solicit 
the  body,  these  equations  become  in  that  case 

A  J+(C  — B)m'«"=0^ 
B-^+(A— C)««"=0 


}>....(.). 


dt 

C^+(B— A)««'  =  Oj 

In  these  equations  A,  B,  C,  are  constant  for  the  same  body,  and 
putting  for  abridgment 

B— C      ^     C— A      ,^    A— B      ^ 

they  become  rf«  =  L  o'  «"  dt,  dot'  =  M  «  «"  dt,  da"  =  Nu  «'  dt. 
Multiplying  these  severally  by  u,  «',  «",  and  putting  ua'co"dt=:d^, 
we  have  «rf«  =  L  df,  «'a«'  =  M  d^,  w"t?»"  =  N  d^,  and  the  in- 
tegrals of  these  are 

u2  30 


234  ELEMENTS    OF    DYNAMICS. 

u"  =  2L  ^  +  a\  u'"  =  2M  ip  +  6«,  «'"»  =  2N  <}.  +  c«  .  .  .  .  (2), 
where  a,  b,  c,  are  what  u,  u',  u"  become  when  /,  which  is  the  in- 
dependent variable  in  the  function  f,  becomes  0  ;  that  is,  those  con- 
stants are  the  initial  rotatory  velocities  about  the  axes. 

Consequently,  substituting  these  values  foruw'co"  in  the  equation 

dt= 7-77-'  w^  shall  have,  for  the  determination  of  *,  correspond- 

ing  to  any  time  t",  in  functions  of  /,  the  differential  equation 

v/K2L<}'-fa2X2M.}.+62)(2N4.  +  c2)f 
Suppose  now  that  an  initial  velocity  a  is  given  to  the  body  about 
one  only  of  the  principal  axes,  then  6  •=  0,  c  =  0,  and  this  expres- 

sion  becomes  dt  =^^^^~^  ■  f^^^r^f^'  ^hat  is,  replacing 
2L^+a®  by  its  value  w',  and  d^  by  its  value— jt^  , 

.      1__        ^" 

^       ^/JiMNf  «»  — a"' 
and  the  integral  of  this  is 

^  ^  ^         *       2a    ^     «-f  a 

.-.  e'"'C.  c2<"'>/mn=    "~"    ....  (3). 
lo-f-a  ^  ^ 

Now  the  constant  C  must  be  determined  so  that  when  /=0,  w  may 
be  =a,  that  is,  the  first  member  must  vanish  for  t=0  ;  hence  e*"^ 
must  =0,  or  C  =  —  co,  consequently,  there  must  always  be  uz=a, 
and  therefore  (2)  always  $=0,  w'=0,  a)"=0  ;  consequently,  as  be- 
fore shown,  the  impressed  velocity  about  one  of  the  principal  axes 
of  rotation  continues  perpetual  and  uniform. 

We  see,  moreover,  from  the  equation  (3),  that  if  the  instanta- 
neous axis  of  rotation  does  not  coincide  with  a  principal  axis  at 
the  commencement  of  motion,  it  cannever  afterwards  coincide  with 
it :  for  if  we  suppose  the  coincidence  to  take  place  at  any  epoch,  and 
that  the  angular  velocity  is  then  a,  then,  measuring  t"  from  that 
epoch,  the  foregoing  equation  must  give  u>=a  when  /=0,  which  re- 
quires, as  shown  above,  that  C=  —  od,  and  therefore  w  must  be  al- 
ways =a,  for  every  value,  positive  or  negative,  of  t. 

(178.)  Let  us  now  see  what  must  be  the  conditions,  in  order  that 
the  instantaneous  axis  of  rotation,  if  it  do  not  accurately  coincide 
with  one  of  the  principal  axes,  may  yet  always  be  very  nearly  co- 
incident. 

Let  us  suppose  the  axis  of  instantaneous  rotation  to  be  nearly  co- 


PRINCIPAL    AXES    OF    ROTATION.  235 

incident  with  the  axis  of  z ;  then  considering  the  angular  motion  to 
be  the  resultant  of  three,  about  the  three  principal  axes,  the  veloci- 
ties  o,  w',  about  the  axes  of  x  and  y,  are,  by  hypothesis,  to  be  very 
small  in  comparison  with  the  angular  velocity  to"  about  the  axis  of 
z,  because  the  body  turns  almost  entirely  about  this  axis.  The  ex- 
pression for  the  sine  of  y,  the  angle  which  the  instantaneous  axis 
makes  with  that  of  z,  is  (p.  222,) 

sm.  y= — ^— ^ ! i— 

Now,  on  account  of  the  smallness  of  both  w  and  co',  the  tliird  of 
the  equations  (1)  p.  233,  becomes  C  -^=0  very  nearly ;  so  that 

«",  the  velocity  round  the  axis  of  z,  is  very  nearly  constant.     Call 
it  w"=w,  then  the  remaining  equations  of  (1)  become 
.  dui      ,-,      „, 

B-^+(A-C)nw=0j 
By  differentiating  the  first  of  these,  we  have 

but,  from  the  second,  —  =  — - — n^ ;   hence,  by  substitution, 

rf.        (A-C)(B-C) 
dt^  +  AB  ""  "-^ 

or,  putting  for  brevity  the  coefficient  of  io=/2, 

^+;»„=o....(2). 

The  integral  of  this  equation  is,  (see  Int.  Calc.  p.  233,) 

1  o 

t-{-c'=-rsm.  — 1 — . 
/  c 

If  at  the  commencement  of  motion,  or  when  t=0,  «  were  accu 

lately  0,  the  constant  c'  would  necessarily  be  0 ;  as  it  is,  however, 

c'  must  be  very  small :  calling  Ic',  k,  we  have 

«=c  sin.  (It-\-k)  .'.  —  =lc  .  cos.  {lt-\-k) ; 

and,  substituting  this  last  expression  in  the  first  of  (1),  we  get  for 
,   ,         ,        ,     Ale  .  cos.  (It4-k) 

«     the  value  W  ==  rrr ^^-^rrr ^. 

n{B  —  C) 
Here  also  we  may  observe  that  if,  at  the  commencement  of  the 
motion,  the  instantaneous  axis  accurately  coincided  with  the  axis 
of  z,  c  would  necessarily  be  0,  for  otherwise  w  and  «'  would  never 


236  ELEMENTS    OF    DYNAMICS. 

be  both  0 ;  so  that  we  again  infer  wliat  has  been  otherwise  proved, 
viz.  that  w  and  u'  if  0  at  the  beginning  are  always  0.  But,  if,  as 
we  here  suppose,  u  and  w'  are  not  accurately  0  at  tlie  commence- 
ment, then  c,  instead  of  being  0,  must  be  very  small ;  consequently, 
if  /  involves  no  impossible  quantity,  w  and  w'  must  always  be  small, 
however  great  /  may  be,  for  be  this  as  great  as  it  may  the  factors 
sin.  (lt-\-k),  cos.  {lt-\-k),  can  never  exceed  unity. 

The  expression  for  /  is  l=^n^  \  — -\  ;  where  A, 

B,  C  are  essentially  positive ;  hence  that  the  expression  may  be 
possible  A  —  C,  B  —  C,  must  be  cither  both  positive  or  both  nega- 
tive ;  that  is,  C,  the  moment  of  inertia  with  respect  to  that  axis 
about  which  the  axis  of  instantaneous  rotation  perpetually  oscillates, 
must  be  either  the  least  or  the  greatest  of  the  three  moments  A,  B,  C. 

If  we  were  to  integrate  (2)  on  the  hypothesis  that  /  is  imaginary, 
or  l^  negative,  the  resulting  expressions  for  to,  w',  would  be  expo- 
nentials, t  entering  as  an  exponent ;  their  values  would,  therefore, 
increase  continually  with  /,  showing  that  the  supposition  of  these 
quantities  continuing  small,  after  the  commencement,  is  inadmis- 
sible. 

We  conclude,  therefore,  that  when  a  body  commences  to  revolve 
about  a  principal  axis,  it  will  perpetually  do  so  :  when  it  happens 
that  any  extraneous  cause  deranges  this  uniform  rotation  a  little,  the 
body  will,  nevertheless,  always  have  the  new  axes  of  rotation,  whicli 
it  perpetually  turns  about  very  near  to  the  original  axis,  provided 
this  happened  to  be  either  the  axis  of  greatest  or  of  least  moment ; 
but  if  it  happened  to  be  the  axis  of  mean  moment,  then,  however 
trifling  the  derangement  may  have  been,  its  effects  will  increase  with 
the  time,  and  the  body  will  depart  altogether  from  its  original  mo- 
tion. We  hence  say  that  the  rotation  about  the  principal  axis  of 
greatest  or  of  least  moment  is  stable,  while  the  rotation  about  the 
axis  of  mean  moment  is  unstable. 


CHAPTER  VII. 

MISCELLANEOUS    DYNAMICAL    PROBLEMS. 

Problem  I. — (179.)  If  a  body  revolve  about  any  centre  of  force 
S,  (fig.  129,)  and  if  the  velocity  at  an  apse  A  is  v,  the  expression 
for  the  sector  ASP,  described  by  the  radius  vector  in  any  time  / 
from  A,  will  be  ASP  =  |  AS  .  v  .  t,  required  the  proof. 

Taking  S  for  the  origin  of  the  rectangular  axes,  and  SA  for  tlie 


MISCELLANEOUS    DVNAMICAL   PROBLEMS.  237 

axis  of  X,  the  components  of  the  velocity  in  the  curve  are  always 

dx      .  dy  .  ,         ,     .  A 

dt  dt'  ^^        ^P^^'        v<^locity  in  the  curve  being  in  a  parallel 

direction  to  the  axis  of  y,  and  being  0  in  the  direction  of  x,  we  have, 
at  this  point,  ^=0,  -^=v;  therefore,  since  (122)  ^'^^yzZjI^^c, 
we  must  have  at  A,  a;t;=AS  .  v=c  .-.  |cf =sector  ASP=i  AS  .v  .  /. 

Problem  II. — To  determine  the  curve,  such  that  if  it  revolve 
about  its  axis  placed  vertically,  with  a  given  angular  velocity,  a 
heavy  ring  at  liberty  to  slide  along  it  shall  remain  wherever  it  is 
placed,  (fig.  130.) 

Let  CBA  be  the  curve,  and  u  the  given  angular  velocity,  and  let 
w  be  the  point  where  the  ring  is  placed :  draw  RN  a  normal  to  the 
curve  and  RM  perpendicular  to  the  axis  •?  also  let  ST  be  a  tangent 
at  R,  and  call  the  angle  NRM,  &c.     The  absolute  velocity  of  R  is 

MR^  <o^ 

MR.  w,  and,  therefore,  the  centrifugal  force  is '- — =MR  .  w^ 

^  MR 

in  the  direction  RR' ;  the  component  of  this,  in  the  direction  RT 
of  the  curve,  is  MR  .  u^  sin.  a,  and  the  component  of  the  force  g  of 
gravity  in  opposition  to  this,  that  is,  in  the  direction  RS,is  gcos.a; 
hence,  in  order  that  the  ring  may  rest,  these  two  forces  must 
balance  each  other,  .•.  MR  .  co^  sin.  a.=.g  cos.  a 

MN  2- 

•■•  ^^  •  "'mr="'  •  ^^=^  •■•  ^^=5' 

that  is,  the  subnormal  is  constant,  and,  hence,  the  curve  is  a  parabola 
with  its  vertex  downward.  Upon  similar  principles  we  may  deter- 
mine the  concave  surface  which  a  fluid  presents  when  a  rotatory 
motion  is  given  to  the  vessel  which  contains  it.  This  surface  is  a 
paraboloid. 

Problem  III. — An  upright  cylinder  standing  on  a  smooth  hori- 
zontal plane  has  a  string  coiled  several  times  round  it  in  the  plane 
of  its  base ;  one  end  is  fixed  to  the  cylinder,  and  to  the  other  is 
attached  a  body  P,  to  which  a  velocity  is  given  in  the  direction  of 
the  string;  to  determine  the  motion,  (fig.  131.) 

Let  V  be  the  progressive  velocity  of  the  cylinder  at  any  instant, 

then  M  being  its  mass,  Mu  will  be  the  impulsive  force  due  to  this 

velocity ;  as  it  acts  at  the  extremity  of  the  radius  r  of  the  base,  and 

in  the  direction  of  a  tangent,  it  will  communicate  to  the  circum- 

Mw  •  r^ 
ference  of  the  cylinder  the  rotatory  velocity  =2v;  therefore, 

the  absolute  velocity  of  every  point  of  the  unwound  string  must,  at 


238  ELEMENTS  OF  DYNAMICS. 

that  instant,  be  3v,  which,  therefore,  expresses  the  correspondinc 
velocity  of  P ;  hence,  at  that  instant,  the  whole  momentum  of  the 
system  is  Mu  +  3Pi;;  but  if  the  given  velocity  originally  commu- 
nicated  to  P  be  I',,  the    momentum  communicated  will  be  Fi\, 

Vv 
.-.  Mv+3  Pu=Prj  .-.  v=  ri— Tu  '   *'^'^'  therefore,  is  the  velocity 

of  the  cylinder  in  progression,  at  any  instant,  and  twice  this  is  the 
rotatory  velocity  of  the  circumference ;  the  velocity  is,  therefore, 
uniform,  so  that  the  motion  of  the  cylinder  is  wholly  due  to  the 
initial  impulse  it  receives  from  P ;  P,  therefore,  never  afterwards 
acts  on  the  cylinder. 

Problem  IV. — A  uniform  straight  rod  AB  (fig.  132,)  is  placed  in 
an  assigned  position  upon  a  smooth  horizontal  plane,  and  one  end 
of  it,  B,  is  drawn  uniformly  along  the  straight  line  CD  with  a  given 
velocity  u  ;  it  is  required  to  find  the  position  of  the  rod  at  any  time, 
and  its  angular  velocity.     See  note  at  page  257. 

Let  G  be  the  centre  of  gravity  of  the  rod,  then  the  uniform 
motion  of  B  along  the  straight  line  CD  may  be  considered  as  the 
consequence  of  an  impulse  at  G,  in  the  direction  GD',  parallel  to 
CD.  As  the  progressive  velocity  thus  generated  is  v,  the  value  of 
the  impulsive  force  is  2a-  v,  a  representing  the  mass  of  half  the 
rod  or  of  AG.  But,  as  this  force  really  acts  at  B  instead  of  G, 
there  would  be  generated  in  addition  to  the  progressive  velocity  a 
uniform  angular  velocity  about  G,  if  B  were  not  constrained  to 
continue  on  the  line  CD  ;  as  it  is  the  angular  motion  must  be  about 
B.  Now  the  same  force  which  applied  at  B  produces  any  angular 
motion  of  the  rod  about  G,  would,  obviously,  if  applied  at  A,  and 
in  an  opposite  direction,  produce  the  same  angular  motion ;  what- 
ever angular  motion,  therefore,  has  place  in  the  present  case,  is  due 
to  the  force  2  av  applied,  at  the  commencement  of  motion,  to  the 
point  A  in  the  direction  AC'.  Calling,  therefore,  the  angle  ABY, 
which  the  rod  makes  with  the  perpendicular  BY  at  the  beginning 
«j,  we  have,  for  the  constant  angular  velocity  about  B, 
du  2ayxBE  2ayx2  «cos.  w, '  3  v 
dt^i{2ay~^  p  ^1'  '^  *^°^'  "^ ' 

hence,  at  any  time  /",  the  rod  will  make  with  BY  an  angle  w  equal 

V 

to  I  <  •  —  cos.  Wj,  and  the  length  of  path  gone  over  by  B  will  have 

been  equal  to  tv,  so  that  the  position  and  angular  velocity  of  the 
rod  at  any  time  is  completely  determined. 

The  curve  traced  out  by  any  point  in  the  moving  rod  is,  ob- 
viously, a  species  of  cycloid,  for  each  point  describes  uniformly 
the  circumference  of  a  circle,  whose  centre  B  is  uniformly  moving 


MISCELLANEOUS    DYNAMICAL   PROBLEMS.  239 

along  a  straight  line ;  that  point  in  the  rod  whose  rotatory  ve- 
locity is  equal  to  the  progressive  velocity  of  B  will  describe  the 
common  cycloid  ;  for  if  with  centre  B  a  circle  be  described  throutrh 
this  point,  and  then  a  tangent  to  it  be  drawn  parallel  to  CD,  this- 
circle,  by  rolling  on  the  tangent  so  that  its  centre,  or  the  point  of 
contact,  may  move  with  the  proposed  velocity  of  B,  will  obviously 
cause  every  point  in  the  circumference  to  revolve  with  that  same 
velocity,  and  thus  the  point  in  question  will  trace  the  path  which 
it  actually  has,  and  Avhich  must,  therefore,  be  the  common  cycloid. 

Problem  V. — Suppose  a  heavy  particle  is  placed  at  a  given 
point  in  a  perfectly  smooth  narrow  tube  of  a  given  length,  and  sup- 
pose the  tube  to  be  whirled  about  one  end  as  a  centre  with  a  given 
angular  velocity  in  a  horizontal  plane  ;  it  is  required  to  determine 
the  velocity  and  direction  of  the  particle  when  it  leaves  the  tube ; 
the  motion  being  solely  generated  by  the  revolving  tube,  (fig.  13.3.) 

Let  SA=a  be  the  tube  in  its  first  position,  and  B  the  place  of  the 
particle.  Call  SB,  b,  and  the  distance  SP  of  the  particle  at  any 
time  t",  r ;  let  also  v  represent  the  uniform  angular  velocity  of  the 
tube,  and  SC  its  position,  when  the  particle  quits  it. 

As  no  centripetal  force  acts  on  the  particle,  its  motion  along  the 
tube  is  entirely  due  to  the  centrifugal  force  rv^  (art.  138),  that  is 

— — =  rv^. 
dp 

Multiplying  this  by  2  dr,  and  integrating,  we  have 

dr^  dr^ 

—  =r^v^+C,  also  0=bv^+C  .-.  —=v^  (r^  —  b')  .  (1)  ; 

dt'^  dr 

hence,  when  r=a,  or  when  the  body  arrives  at  the  mouth  C  of  the 

tube,  its  velocity  in  the  direction  CE  of  the  tube  is  v  v/{«" — b^  \ 

also  its  velocity  in  the  direction  CD,  perpendicular  to  this  is  va ; 

hence  its  velocity  in  its  path  at  that  point  is  the  component  of  these, 

viz.  V  'y{2a^  —  b^\  and  for  the  angle  ECF,  which  the  direction 

makes  with  the  tube,  we  have 

„„„     CD  va  a 

tan.Z-ECF=^=^^^^,_^,,  = -^^-^-. 

This  angle,  added  to  the  angle  S,  will  give  the  position  of  CF, 
with  respect  to  AS.  If  T  represent  the  time  in  which  the  tube 
passes  from  the  position  SA  to  SC,  then  since  v  is  the  angle 
described  in  one  second,  Tv  will  express  the  angle  S,  and  to  find 
T,  we  have,  from  the  equation  (1), 

1  r-{-^\r'  —  b»\      ^      1     ,       a+^{a-'^b'\^ 


t=-\og.  .^^^^ ^  .-.  T=-xlog 

a+^\a^—b 


V         "  b 


so  that  the  angle  S  is  expressed  by  log. 


240  KLEMENT8  OF  DYNAMICS. 

Pnoni-EM  VI. — Two  bodies  P  and  Q  (fijr-  l^^O  ^re  plarod  on  a 
smootli  horizontal  plane,  and  connected  by  a  perfectly  flexible  and 
inextensible  thread,  which  passes  freely  through  a  small  rinj^  R  in 
the  plane  ;  a  given  velocity  is  communicated  to  1*  in  a  given  di- 
rection :  it  is  required  to  find  the  equation  of  the  curve  described  by 
P  on  the  plane,  the  length  of  the  thread  being  indefinite. 

Call  the  radius  vector  RP,  r,  and  let  P,  Q  represent  the  masses 

d'r 
of  the  bodies,  then  -r-j  being  the  acceleration  of  Q,  the  motive 

d'r 
force  of  Q  must  be  Q  —j-^ ;  hence  P  must  be  drawn  towards  the 

centre  R  by  this  motive  force ;  the  accelerative  force  on  P  is  there- 

Q(i*  r 
fore  —  — r-  which  may  be   considered  as  the  centripetal  force  at  R 
P  (If*  ^  * 

which  retains  P  in  its  orbit. 

rf'r  . 
Now  we  know  that  the  paracentric  force  — r-j  is    equal    to   the 

difTerence  between  the  centripetal  and  the  centrifugal  force  — ,  (see 

art.  138,)  therefore  5^-f  1^^=|  ....  (1). 

Multiplying  by  2  dr  and  integrating, 

O  dr^  1  1  c^      r'^ r  * 

or,  substituting  for  the  square  of  the  paracentric  velocity  its  value 

/  5  f^ r    s  >  I  P_1_Q"      1 

at  page  170,  we  have  — ^ —  =     I    „   ^    •  J  .  «  being  the 

angle  between  r  and  any  fixed  line  RX.     Consequently, 

.  _     |P  +  Q  c,dr   

-^     P     '  rx/lr*  — c,''!' 

JP-fQ  r 

— — — .  sec— J  —  .  .  .  .  (3),  the 

polar  equation  of  the  orbit. 

The  constants  c^  and  c^  are  readily  determined  from  the  initial 
conditions  of  the  motion ;  thus  let  v,  be  the  initial  velocity,  and  o 
the  angle  which  its  direction  makes  with  the  original  position  of  the 
string  r,  and  let  the  value  of  r  be  then  a ;  the  initial  velocity  in  the 
direction  of  the  radius  vector  will  consequently  be  u  cos.  a,  hence, 

(If 
putting  this  for  -y-  in  the  equation  (2),  we  have,  for  the  determina- 
tion of  Cj  ,  this  equation,  observing  that  c,  which  denotes  double  the 


MISCELLANEOLS    DYNAMICAL    PROBLEMS.  241 

sectorial  area  described  about  R  in  one  second,  and  which  is  con- 
stant, is  expressed  by  a  v^  sin.  a, 

.  .  ,      .         r        1    .V,         1         1         1      ,P  +  Qcos.«a 

which  ffives  for  —  the  value  — = — v" — :^^ . 

^  Cj  c^      a  Psin.^a 

To  determine  c^  let  the  radius  vector  coincide  with  the  assumed 
line  RX  at  the  commencement;  then  u=0  when  r=a,  and  con- 
sequently, 


-J 


P  +  Q  ,       P  +  QcOS.^a 

— — —  .  sec.— 1^/- 


P  sin.^  a 

Problem  VII, — A  given  weight  W  (fig.  135,)  hangs  at  the  middle 
point  71  of  a  string  PCnC'P'  passing  over  the  pulleys  C,  C,  in  the 
same  horizontal  line,  drawing  up  the  two  equal  weights  P,  P',  hang- 
ing at  the  extremities  of  the  cord  ;  to  determine  the  motion  of  W. 

Draw  nD  perpendicular  to  the  horizontal  line  CC'  then  CD=C'D 

Let  CD=a,  Dn=x,  Cn=y,  then  —  =cos.  CnD=cos.  C'nD 

y 

Now  to  the  point  n  there  are  applied  the  force  P,  in  the  direction 
nC,  the  force  P'  in  the  direction  nC',  and  the  force  W  in  the  di- 
rection nW. 

The  actual,  or  effective  forces,  in  these  directions,  are  respectively 

—  .  -7^ ,  —  .  —r^  ,  —  .  -; —  ,  and  of  these  it  must  be  observed 
g      dt^     g      dt^      g      dP 

that  the  first  two  have  contrary  directions  to  the  corresponding  ap- 
plied forces  ;  for  the  applied  force  P  acts  from  n  towards  C,  where- 
as the  actual  force  is  from  C  towards  n,  since  W  descends  ;  conse- 
quently, taking  the  effective  forces  in  opposite  directions  to  those 
which  they  really  have,  there  must,  by  the  principle  of  D'Alembcrt, 
be  an  equilibrium  among  the  forces 

^  g  '   dV'  '      ^  g'    dV^'  g  '  dt^' 

acting  at  n,  in  the  directions  ??C,  n(^'  nW.  Consequently  the  verti- 
cal components  of  these  must  equilibrate,  that  is, 

^^^^  g'  dt^->  2/  -^^       g'  dt^  "-^^^' 
The  relation  between  x  and  y  is  y"^  =a^-\-x^  .'.  —  =  -r^  , ,  ,  (2) 

X 

therefore,  substituting  this  value  of  —  ,  dividing  by  2P,  and  putting 

y 

W 

for  abridgment  — —  =7n,  we  have 

At    1 

X  31 


242  ELEMENTS  OF  DYNAMICS. 

"'l^^'^'^'d'^y^^  (r«</:r  — f/j/) (3); 

and,  by  integration,  im  —  +i    i7^=S  {nix  —  y)  +  c  .     .  .  (3) ; 

dy^        x^     dx~ 
or,  since  in  virtue  of  the  condition  (2),  —r-  =  —  .  —r-^  this  equa- 

^  ^    rfr         y^      dt^  ^ 

dx^        „  2  s;  (mx — v)+2c 
tion  gives  — r—  =v   — ^^r-; — -t ; 

which  expresses  the  square  of  the  velocity  of  W.  The  determina- 
tion of  c  depends  upon  the  initial  position,  when  the  velocity  is  0 ; 
thus  c^g  (?/i  —  mx^),  where  x■^^  and  y^  are  the  values  of  a;  and  y, 
at  the  commencement  of  motion. 

To  find  when  the  velocity  is  again  0  we  have/rom  (3),  by  intro- 
ducing this  value  of  c,  the  equation  g  [mx  — y)-\-g  (^/i  —  wxJ^O  ; 
or,  substituting  for  y  its  va\\iey/\a'^-\-x^],  and  reducing,  we  have 
the  quadratic  equation 

(1  — m^)  x~ —  (2  my  I —  2m^x-i)  x  —  (y^ — mx.^)--\-a^=0. 
One  of  the  roots  of  this  equation  is  by  hypothesis,  x=x^,  and,  if 
we  represent  the  other  by  x^,  we  have,  by  the  theory  of  equations, 
_  2m  (y,—mx,).       _  2my,~(m'^+l)  a^i 

If  m=l,  that  is,  if  W=2P,  W  will  continually  descend,  and  never 
become  stationary,  as  remarked  at  page  30,  and  the  same  will  oi 
course  happen  if  W  >  2P  ;  hence,  for  the  system  to  become  sta- 
tionary at  any  time  after  the  commencement  of  motion,  W  must  be 
less  than  2P,  and  the  distance  below  the  horizontal  line,  at  which 
this  will  take  place,  will  be  given  by  the  above  value  of  x^ ,  so  that 
the  weight  W  will  move  continually  backward  and  forward  between 
the  two  points  x^,  x^.  Whenever  the  relation  between  2P  and  AV 
is  such  as  to  render  x^  negative,  it  will  be  impossible  for  the  system 
to  become  stationary  at  any  time  after  the  commencement. 
This  problem  may  be  solved  otherwise  as  follows  : 
Since  the  components  of  the  forces  P,  P'  in  7iC,  wC  are  together 

X 

equal  to  2  P  — ,  a  portion  of  W,  equal  to  this,  is  expended  in  balanc- 
ing the  equal  weights  P,  P',  hence  the  moving  force  is  only  W  — 

X 

2P  — ,  and  this,  divided  by  the  inertia  opposed  to  motion  at  the 

d'^x 
point  n,  must  give  the  acceleration  -j-^  of  W.     The  inertia  due  to 

W 

W  is  its  own  mass  —  ;  the  inertia  of  the  other  bodies  must  be  ex- 

g 
pressed  by  such  a  mass  placed  at  n,  or  incorporated  with  W,  as 


MISCELLANEOUS  DYNAMICAL    PROBLEMS.  243 

when  multiplied  by  the  acceleration  of  W  will  give  the  same  mo- 
tive force  in  the  vertical  direction  that  the  real  bodies  P,  P',  acting- 
along  nC,  nC  have  in  the  vertical  direction.     The  motive  force  of 

P  +  P',  or  2P,  in  the  direction  nW  is  2  —  .  —  .  — •" ;  to  determine 

g      y      dt^ 

therefore  what  mass  M,  having  the  acceleration  -r-^  ,  must  have  the 

same  motive  force,  we  have 

M-^  =  2^  ^  i^.M='>^  ^i'}!y^£-^^ 

dt'  g      y       dl'  ^  S  '   y      dt^   '    di" 

Dividing,  therefore,  the  motive  force  W — 2  P  —  by  the  whole  in- 

y 

d^x 
ertia  at  n,  we  have  for  the  acceleration  — —  the  equation 

dp 

g  (W— 3  P  — ) 

^  ^  y  ^  d^x 


^^'^^^J^'dF"^dF-> 

therefore,  putting  as  before 

.      W  dy  .      X  d^x      d^y  ^ 

m  for ,  and  -^  lor  — ,  g(mdx  —  dy)  =  m  — i  dun 

2p  dx         y^^  ^^  dt^  ^  dt^    '^' 

which  is  the  same  as  equation  (3)  before  determined. 

Problem  VIII. — Two  given  bodies  P,  Q  (fig.  136,)  are  connect- 
ed together  by  a  string,  which  passes  over  a  fixed  pulley  at  a  given 
distance  from  a  smooth  horizontal  plane.  It  is  required  to  deter- 
mine the  circumstances  of  the  motion  when  P  is  drawn  alono-  the 
plane  by  the  descending  weight  Q. 

Let  the  velocity  of  Q  be  v,  and  the  velocity  of  P,  M ;  then,  call- 
ing PB,  X,  and  PC,  s,  CQ,  y,  and  the  angle  P,  a,  we  have 
V       ds  X 

u       dx  '  s  ' 

hence  the   actual  motive  force  of  P,  in  the  direction  PC,  and  of 
which  the  force  in  PB  is  the  component,  is 
du    F  1       _  du    P    u 

dt     g      cos.  o       dt     g     V  '  '  '  '  ^  '' 
the  impressed  force  on  P  is  gravity  and  the  resistance  of  the  plane 
so  that  no  moving  force  is  impressed. 

Again,  the  actual  force  of  Q  is  —. —  — ,    and    the    impressed 

dt       g 

force    is   the  weight  Q.     Hence,  by  D'Alembert's  principle,  the 

force  (1,)  acting  in  the  direction  CP,  equilibrates  the  force 


244  ELEMENTS   OF    DYNAMICS. 

acting  in  the  direction  CB,  that  is, 

at      V  °       at 

.-.  P  u  du-\-Q,  V  dv=Q  g  V  dt=Q,  g  dy 
and  integrating  this  equation,  P  Ji'^  +  Q  y^=2  Q  g  (y — c),  c  being 
the  value  of  y  at  the  commencement  of  motion,  therefore,  since 

we  have  v'^=: ~ j^ for  the  square  of  the  velocity  of  Q, 

P  s^+Q  x^ 

»  and  y  being  known  in  terms  of  x  from  the  known  length  of  CB, 
and  of  the  string ;  and  the  velocity  of  P  is  then  found  from  the  pre- 
ceding equation. 

The  problem  may  be  solved  otherwise  as  follows : 
Having  found,  as  before,  that  the  motive  force  of  P  is 
du    F      u 
dt  '  g  '    V  * 
we  have,  for  the  determination  of  the  mass  M,  Avhich,  when  accele- 
rated with  Q  has  the  same  motive  force,  or  offers  the  same  resistance 
to  motion,  the  equation 

-^  dv      du    F     u       ,,      P     M     du 
dt      dt     g     V  g     V      dv 

Q 
hence  the  whole  inertia  of  the  system  being  M  -| — ,    we  have  for 

the  acceleration  of  Q 

Q  Qgv  dv  ,  ,^„  s 

—  ^  =^T  (P^Se  123.) 


M  +  —      Qv  +F  u  —  ^ 

g  dv 

Consequently  Q^  dy=Q  v  dv-^F  u  du,  as  before. 

Problem  IX. — A  perfectly  flexible  chain  is  wound  round  a  cylin- 
der, supported  with  its  axis  parallel  to  the  horizon.  Then  the 
weight  and  dimensions  of  the  cylinder  being  given,  as  also  the 
weight  and  length  of  the  chain  :  it  is  required  to  determine  the  time 
in  which  the  chain,  impelled  by  the  force  of  gravity,  will  unwind 
itself;  a  given  length  being  unwound  at  the  commencement  of  the 
motion. 

The  moving  force  here  always  acting  is  due  to  that  part  of  the 
chain  which  hangs  down,  and  the  resistance  to  be  overcome  is  the 
rotatory  inertia  of  the  cylinder,  and  of  the  mass  of  chain  which  en- 


MISCELLANEOUS    DYNAMICAL    PROBLEMS.  245 

velopes  it.  As,  (page  189,)  the  square  of  the  radius  of  gyration  of 
the  cylinder  is  gR'-',  and  that  of  a  mere  circumference  of  the  same 
radius  R,  R^,  it  follows  that  the  system  would  ofler  the  same  resist- 
ance to  rotation,  if  for  the  cylinder  we  substitute  an  indefinitely 
slender  ring  of  the  same  radius,  containing  half  the  mass  of  the  cy- 
linder, and  the  whole  mass  of  the  unwound  chain. 

Let  then  2W  denote  the  whole  weight  of  the  cylinder,  and  w 
that  of  the  chain  ;  let  the  length  of  the  chain  be  /,  a  the  length  hang- 
ing down  at  the  commencement  of  the  motion,  and  x  the  length 
hanging  down  at  any  time  t" .     Then  the  weight  of  any  part  being 

as  its  length,  we  have  I  :  x  : :  iv  :  —r-,  which  expresses  the  moving 

force  acting  at  the  extremity  of  the  radius  at  any  time  t"  ;  the  value 

wx 
of  this  force  so  acting  to  turn  the  system  is  R— r-j  and  this,  divided 

by  the  moment  of  inertia,  will  give  the  angular  acceleration  (158), 
and  consequently  the  acceleration  of  the  extremity  of  the  radius,  or 
of  the  descending  chain,  is 

„„  wx       „„  ,W+?t'      ,       .     d^x              wx 
R2  — r-  -T-  R2  ( — -?— ),  that  IS,  -~=  ^-ttttt r- 

I  g  dt^    *  i{yi+w) 

dx 
Multiplying  this  by  2-^,  we  have 

2  d^^  X      dx  wx       dx 

dt^     '  ~dt^    °   l{W+zv)  dt' 
This,  multiplied  by  dt  and  integrated,  gives 

dx^       „  tvx^  ^ 

dP  ^    /(W+w)   ^ 

To  determine  C  we  have  the  condition  v=0  when  x=a, 

w  a^ 

■■■  »=^7(W+1^+  ^  • 

hence,  subtracting  this  equation  from  the  former, 

w  Ix^—a^)  dx         ,   w  (x^— a^) 


l{yj  +  w)  dt      ^  '^  /(W  +  w;)  ^' 

consequently 

dt-  ^/  \ \  TJ^Zr^a-y 

the  integral  of  which  is  (Int.  Calc.  art.  18), 

which  is  the  number  of  seconds  occupied  in  unwinding  the  length 
X  —  a,  the  part  a  being  unwound  at  the  beginning  ;  and  when  x  =  1 
x2 


246  ELEMENTS  OF  DYNAMICS. 

we  have,  for  the  number  of  seconds  occupied  in  unwinding  the 
whole  chain, 


Problem  X. — To  a  point  in  the  circumference  of  the  base  of  an 
upright  cylinder,  standing  on  a  smooth  horizontal  plane,  is  fastened 
one  extremity  of  a  string,  and  to  the  other  a  weight  P.  Now  a 
given  velocity  is  communicated  to  P  in  a  direction  perpendicular  to 
the  string:  to  determine  the  circumstances  of  the  motion  (fig.  137). 

Let  A  be  the  point  to  which  the  string  is  fastened,  and  B  the  point 
of  contact  at  any  time  t" ',  let  also  AOB=o,  BP=:z,  AO==o,  ten- 
sion of  the  string  =T,  then,  taking  ON,  NP  for  the  rectangular 
co-ordinates  of  P  at  the  time  t" ,  we  have,  since  the  acceleration  of 

T 

P  in  the  direction  PB  is  — ,  these  equations  of  its  motion  in  the 

directions  of  the  co-ordinates,  viz. 

cPx  T   .  d^y  T 

^  =  -p«"-"'^  =  -p- COS.  «....(!), 

T 

the  negative  signs  being  used  because  the  accelerative  force  — ,  being 

in  the  direction  PB,  tends  to  diminish  the  co-ordinates. 

Multiplying  the  first  of  these  equations  by  cos.  co,  and  the  second 

by  sin.  w,  and  subtracting,  we  have 

d^'x        .        d^y     ^ 

cos.  u>  —r- sm.  to  -—-^=0  .  .  .  .  (2 ). 

dt''  df'  ^  ^ 

Again,  the  angular  velocity  of  the  string,  since  OBN'=o,  is  — ; 

and,  therefore,  the  absolute  velocity  of  P  in  direction  PQ,  of  its 

path,  is  z  — ,  and  therefore  its  components,  in  the  directions  of  the 

co-ordinates,  are  z  cos.  w  .  — ,  and  —  z  sin.  w  -j-,  consequently,  by 

differentiating, 

d^x     dz  doi  .        dJ^  d^a 

—r-r=-rCOS.  CO  —^ jjsm.co  -7- — ^zcos.  «^—. 

rf/2       dt  dt  dp  dP 

d^y  dz   .        du>  doi"  .        d^a 

—rr= r-  sm.  co  -; z  cos.  co  -7-  —  z  sm.  10  -r-r-' 

dp  dt  dt  dt''  dp 

Substituting  these  in  the  equation  (2),  there  results 

dz    fZco  rf^co  dz  (?^co         da 

di'di'^^'dp'^    ''  l'^~'dP~dt' 


MISCELLANEOUS    DYNAMICAL    PROBLEMS.  247 

and  inteerratiiiff  each  side,  W.— =  W. -t- ,•.  — =-— -,  the  anofular 

z         °  dt        2      dt  ° 

velocity  of  the  string  at  any  time  t".     To  determine  c  let  v^  be  the 
initial  velocity  of  P  when  z:=6,  then  the  initial  angular  velocity  is 

V         C  V 

-T^=y-  .'.  c=Uj ;  consequently,  —  =  angular  velocity  of  the  string, 

^1=  absolute  velocity  of  P;  hence,  P  continually  moves  with  its 

initial  velocity.     As  the  centrifugal  force  of  P  is  that  to  which  the 

V  2      Pv  2 
tension  of  the  string  is  due,  we  have,  T=P  —  =j — ^—,  I  being  the 

whole  length  ABP  of  the  string. 

Problem  XI. — To  determine  the  path  of  a  projectile  in  a  resist- 
ing medium. 

Let  R  represent  the  resistance  of  the  medium  in  opposition  to 
the  motion  of  the  body ;  then  the  forces  acting  upon  it,  in  the  di- 
rections of  its  horizontal  and  vertical  co-ordinates,  are 
d^_  dx  d^y  _  dy 

dP~  ds'  dt"~ ^  ds' 

which  are  the  equations  of  the  motion ;  and  by  means  of  which, 
when  the  law  of  resistance  is  known,  the  nature  of  the  trajectory 
may  be  determined.  The  law  generally  received  is,  that  the  re- 
sistance varies  as  the  square  of  the  velocity ;  assuming,  therefore, 
this  to  be  the  case,  the  foregoing  equations  are 

d"^  X  dx  d^y  „  c?V  •  ds' 

— — =  —  mv'^  ^-,  -Tfr  =  —  ff  —  mv^  -r,  or,  smce  1;^=  __ 
dt^  ds     dt^  ^  ds  dt^ 

the  equations  are  the  same  as 

d^x_  ds^    dx  d^y_  ds"    dy 

'dF~~^di^'  d^^'di^         ^~^dP  \F* 

As  the  second  members  of  each  of  these  equations  contain  only 

first  differential  co-efficients,  and  as  the  values  of  these  co-efficients 

remain  the  same,  however  we   change  the  independent  variable, 

(Biff.  Calc.  p.  99,)  the  equations  may  be  written 

d^x 

d^x  ds     dx  d^y  ds    dy     ^  ^       dt 

-dF^-'^Tt  •  W  IF^-^-'^dt'  di  '  ^'^  •••^=-^^* 

dt 

Integrating  this  equation  we  have 

.       dx  ,  _,      rfa? 

log.^=-m.-fC.-.-^=c6— ....(2), 

e  being  the  base  of  the  hyperbolic  system  of  logarithms,  and  c  the 
number  whose  logarithm  is  C. 

The  determination  of  c  will  depend  upon  the  initial  conditions 


248  ELEMENTS  OF  DYNAMICS. 

of  the  motion ;  let  the  initial  velocity  be  v^,  and  a  the  angle  it 
makes  with  the  axis  of  x.     The  component  of  v^,  according  to  this 

dx 
axis,  is     VjCos.a,     and  this  same  component  is  what  -j-  becomes 

when  s=0,  that  is  to  say  v  cos.  a=ce°=c;  hence,  substituting 

dx 
this  value  of  c  in  the  preceding  equation,  we  have^=Vj  cos.  o 

c— "^  ....  (3). 

dy 
Again:  multiplying  the  first  of  equations  (1)  by  -^,  the  second 

by  -j-,  and  taking  their  difference,  we  have 

d^x    dy     d^y    dx_    dx 
IF  '  dt~~dF  '  Tf^H' 

d-^ 
(It*  doc        ddc 

dividing  each  side  by  ~r  the  result  will  be  —  —  •  =^; 

or,  putting  y'  for  -^,  —  —  .  ^  =  g,  an  equation  containing  only 

first  diflferential  co-efficients,  and  which  we  may,  therefore,  write 
thus  (Biff.  Calc.  p.  99) 

^(dx)(dy')  =  g(dt-)', 
the  parenthesis  intimating  that  the  independent  variable  is  arbitraiy. 

Also,  from  equation    (3),  (dt^)  = —^ ^2^5  hence,  by  sub- 

stitution,  — v^-  cos.  ^  »  e-2""  —^  =  g  .  .  .  .  (4). 

Now  y^  =  v/1  -f  (2/'2)  .  • .  (dx)  =     .J  ^^    -,  substituting 
this  value  of  (dx)  in  (4)  and  omitting  the  parentheses,  we  shall 

p-  g2ms  ^g 

have  the  equation  Vl  -\-  y'^  •  dy'  = r— , 

•^  "^  v^  cos.^o 

of  which  the  integral  is 

y'  VT+Y^  +  log.  (2/'  +  vT+y^)  =  c -  ^T^iX "  *  * ^^^' 

dy 
In  order  to  determine  C  we  must  revert  to  the  value  of?/'  or  -^, 

at  the  commencement  of  the  motion ;  this  value  is  y'  =  tan.  a,  cor- 
responding to  which  we  also  have  a;  =  0, 2/  =  0,  5  =  0; 


MISCELLANEOUS  DYNAMICAL  PROBLEMS.  249 

tan.  a  V  1  +  tan.'^o  + 


log.  (tan.  a  +  v^l  +  tan.  ^a)+- 


J 


hence  C 

t  Inor.  ftan.  n.  -4-  V   I  -1-  Ian.  "n.)-*- 

m  V/  COS.^( 
1 

therefore  the  value  of  C  in  the  foregoing  equation  may  be  regarded 
as  known. 

To  determine  a  differential  equation  between  x  and  y  we  may 
eliminate  e^™*  by  means  of  equation  (4)  and  (5) ;  we  shall  thus 
have 

dy: . 


dx-- 


m  ly'^i  +  2/"  +  log.  (2/'  +  v/  1  +  2/'^)  — C! 


also,  since  dy  =  y'dx, 

.   dy==  y^y 


m  \y'^\  +  y'^  +  log.  (y'  +  >/  1  +  y'^)  —  C  \ 
These  equations  are  too  complicated  to  admit  of  integration  in 
finite  terms,  otherwise  we  might  now  obtain  the  values  of  x  and  y 
in  functions  of  ?/',  and  then,  eliminating  y',  we  should  have  a  single 
equation  in  x  and  y  which  would  be  the  equation  of  the  path  sought. 
As  it  is,  we  can  only  obtain  an  approximation  to  the  form  of  the 
curve  described,  which  approximation  may,  however,  be  carried  to 
any  degree  of  exactness.  For  the  method  of  effecting  the  actual 
construction  of  the  trajectary  the  student  may  refer  to  Mr.  Barlow's 
Mechanics,  in  the  Encyclopoedia  Metropolitana,  or  to  Venturoli's 
Mechanics,  and  for  the  general  theory  of  motion  in  a  resisting  me- 
dium, he  may  consult  the  second  book  of  Mr.  Whewell's  Dynamics. 
We  shall  here  terminate  this  miscellaneous  collection  of  prob- 
lems, and  must  refer  the  student,  for  a  more  extensive  variety,  to 
the  Ladies'  and  Gentlemen's  Diaries,  and  to  Leybourn's  Mathe- 
matical Repository,  works  which  cannot  be  too  strongly  recom- 
mended to  the  attention  of  the  mathematical  student,  and  to  which 
our  obligations  are  due  for  several  of  the  foregoing  examples. 


32 


250  ELEMENTS   OF    DYNAMICS, 


NOTE. 

Page  21. 

PoissorCs  Proof  of  the  Parallelogram  of  Forces. 

(180.)  When  two  equal  forces  act  on  a  point  according  to  differ- 
ent directions,  their  resultant,  whatever  it  be  in  intensity,  must  ne- 
cessarily bisect  the  angle  between  these  directions,  as  shown  at  art. 
(7) ;  and  to  determine  the  intensity  of  this  resultant,  M.  Poisson  pro- 
ceeds as  follows : 

Let  mA,  mE  (fig.  138,)  be  the  directions  of  the  components, 

whose  common  value  call  P;  also  let  2a,'  represent  the  angle  AtwB, 

then  mC  being  the  direction  of  the  resultant,  we  shall  have  AmC 

=BmC^a:.     The  intensity  of  this  resultant  can  depend  only  on 

the  quantities  P  and  x,  of  which,  therefore,  it  is  some  unknown 

function.     Representing,  then,  by  R  the  value  of  the  resultant,  we 

shall  have  R=y"(P,  .r).     In  this  equation  R  and  P  are  the  only 

quantities  of  which  the  numerical  value  varies  with  the  unit  of  force 

■p 

that  may  have  been  chosen  ;  their  ratio  r—  is  independent  of  this 

P  \ 

unit ;  whence  we  may  conclude  that  it  must  be  simply  a  function 
of  X,  and  consequently  that  the  function  /  (P,  x)  is  of  the  form 
P  .  ^x.     We  therefore  have 

R=P  .  ^x, 
aud  the  question  is  reduced  to  the  determination  of  the  form  of  the 
function  ^x. 

In  order  to  this,  let  us  draAV  arbitrarily  through  the  point  m,  the 
four  lines  mA',  /nA",  mB',  wB"  ;  suppose  the  four  angles  A'mA, 
A"mA,  B'mB,  B"mB,  equal  among  themselves,  and  represent  each 
of  them  by  z  ;  this  done,  decompose  the  force  P,  directed  accord- 
ing to  mA,  into  two  equal  forces,  directed  according  to  mA'  and 
mA",  that  is  to  say,  regard  this  force  P  as  the  resultant  of  two  equal 
forces  whose  value  is  unknown,  and  which  acts  in  the  given  direc- 
tions mA',  mA".  Representing  the  common  value  of  these  compo- 
nents by  Q,  we  shall  have 

P=Q  .   ^2, 

for  there  ought  to  exist  among  the  quantities  P,  Q,  and  z,  the  same 
relation  as  among  the  quantities  R,  P,  and  x.  Decompose  now  the 
same  force  P,  acting  in  the  direction  mB,  into  two  forces  Q,  acting 


NOTE,  251 

in  the  directions  wB'  and  m'B"  ;  the  two  forces  P  are  thus  replaced 
by  the  four  forces  Q ;  the  resultant  of  these  must,  therefore,  coincide 
in  magnitude  and  direction  with  the  force  R,  Avhich  is  the  resultant 
of  the  two  forces  P.  Now,  calling  Q'  the  resultant  of  the  two 
forces  Q,  acting  in  the  directions  mA'  and  mB',  and  observing  that 
A.'mC=B')nC=x  —  z,  this  force  Q'  will  take  the  direction  mC, 
and  we  shall  have 

Q'=Q.^{x—z). 
In  like  manner,  the  resultant  of  the  two  other  forces  Q,  which  act 
in  the  directions  mA."  and  mB",  will  take  the  direction  mC,  since 
this  line  divides  the  angle  A"?nB"  into  two  equal  parts,  and  because 
A"mC=B"mC=x+z,  we  shall  have 

Q"  =  Q  .  t  (x+z), 
Q"  representing  this  resultant.     The  two  forces  Q'  and  Q"  being 
directed  according  to  the  same  line  mC,  their  resultant,  which  is 
also  that  of  the  four  forces  Q,  will  be  equal  to  their  sum ;  we  must 
therefore  have 

R=Q'  +  Q" 
but  we  already  have  R=P  •  fX^Q,  '  ^z  •  ^x  ;  substituting  then  this 
value  of  R  and  those  of  Q'  and  Q"  above,  and  then  suppressing  the 
common  factor  Q,  there  results 

^x  '  ^z=^  {x  —  z)-\-^{x-{-z). 

This  is  the  equation  which  we  must  now  solve  in  order  to  obtain 
the  value  of  ^x,  or,  which  amounts  to  the  same  thing,  that  of  qiz. 
This  is  effected  in  a  very  simple  manner  by  the  following  considera- 
tions. 

Let  us  develope  f  (x  —  z)  and  q>  [x-\-z),  according  to  the  powers 
ofz,  by  means  of  Taylor's  theorem;  let  us  substitute  these  two  se- 
ries in  our  equation,  and  then  divide  all  its  terms  by  ^,  we  shall 
thus  have 

^  ^    ^  ^x  dx^       2  ^  ^x  dx*     2-S'4^         ^ 

Now  as  ^z  ought  not  to  contain  x,  x  cannot  enter  into  the  coet 
ficients 

d^  ^x       d*  ^x 
^x  dx^  '  fx  dx*  ' 
all  these  quantities,  therefore,  must  be  constants,  that  is,  indepen- 
dent of  the  variables  x  and  z.     Let  b  be  the  value  of  the  first,  we 
have 

d^  AX     , 

whence,  by  successive  differentiation,  we  have 
rf*  ^x       ,  d^  AX      -  d*  AX 


252  ELEMENTS   OF    DYNAMICS. 

d^  ^x  ,d^^o:  ,,         d^  ^x       ,, 

dx^  dx^                   ^x  dx^ 

&c.  &.C.                         &c. 
and,  consequently, 

*z=2U4-  ■ 1 f-&c.L 

or  else  in  replacing  b  by  another  constant  — a'  which  is  allowable, 
a^z\      a*  z*  a^  z^  ,  «     , 

We  recognise  the  series  within  the  parenthesis  to  be  the  de- 
vdopement  of  cos.  az  ;  then  fz=2  cos.  az,  and  putting  x  in  place 
of  z 

^x=2  cos.  ax 
.'.  R=2  P  cos.  ax. 
To  determine  the  quantity  a,  which  we  know  to  be  independent 
of  X,  we  may  observe  that  when  .r=90°  the  two  forces  P  are  di- 
rectly opposite ;  their  resultant  R  is  then  0  ;  so  that  we  must  have 

cos.  (a-90°)=0; 
which  requires  that  a  be  an  uneven  whole  number.     This  whole 
number  must  be  1  ;  for  if  we  had  a  >  1,  for  example  a=S,  the  re- 

90° 
sultant  R  would  become  0  for  x  =:— —  the  two  forces  P  would  then 

o 

equilibrate  without  being  opposite,  which  is  impossible.     We  shall 
have,  therefore, 

R=2  P*  cos.  X. 
This  result  establishes  the  property  otherwise  deduced  in  art.  11, 
viz.  that  any  two  equal  forces  have  for  their  resultant  the  diagonal 
of  the  rhombus  constructed  on  the  straight  lines  which  represent 
them  in  magnitude  and  direction,  and  the  property  may  now  be 
generalized  as  in  the  text ;  the  equation  above  deduced  being  the 
only  thing  connected  with  the  general  theorem  of  the  parallelogram 
of  forces  of  difficulty  to  establish. 


Fa0o  ?jS. 


(  253  ) 
SOLUTIONS  AND  NOTES  BY  THE  EDITOR. 


Problem  XI. — Page  92. — It  will  here  be  convenient  to  use  fig. 
49,  (plate  p.  90,)  which  is  used  in  the  solution  of  prob.  Ill,  p.  85, 
observing  that  in  the  present  question  AC=CB  and  .-.  that  the  per- 
pendicular CE  falls  upon  the  middle  of  AB  and  bisects  the  angle 
ACB,  which  is  here  denoted  by  e.     Then,  as  in  the  problem  cited, 

CF 

we  shall  have  P  .  CE=W  •  -— -,  (1),  by  using   W  instead  of  his 

2P ;  but  CE=CB  •  cos.  -,  and  CF=CB  .  cos.  6,  .-.  by  substitu- 

tion  and  reduction  (1)  becomes  2  P  cos.  -  =  W  cos.  9,  (2) ;  but  (see 

9 

Young's  Trigonometry,  p.  37,)*  we  have  cos.  e=2  cos.=  - —  1  .-. 

0      P         el 

by  subst.  and  reduc.  (2)  becomes  cos.^  — —  ^r^  cos.  — =  —  ,  hence  by 

9  V  P''  1     - 

quadratics  we  have  cos.-=  — ^  ±(^^^+  ^T  as  required.! 

Note. — It  may  be  well  to  observe  that  the  sign  -f  is  to  be  used 

Q 

before  the  radical  in  the  value  of  cos.  —  when  the  rod  lies  as  in  the 

figure  ;  but  the  sign  —  is  to  be  used  when  it  lies  in  an  opposite  di- 
rection. 


Problem  XII. — Let  us  imagine  the  cone  to  be  placed  as  in  the 
problem.  Let  P  denote  the  force  sought,  and/)  the  perpendicular 
from  the  edge  on  which  the  cone  stands  to  the  line  of  its  direction ; 
regarding  the  edge  as  a  fixed  point  round  which  the  cone  can  turn 
freely  in  a  vertical  plane  passing  through  its  axes.  Now  by  putting 
284=:W,  and  p'=:  the  perpendicular  from  the  edge  to  a  vertical  line 
passing  through  the  centre  of  gravity  of  the  cone,  we  shall  have  as 
in  the  last  problem,  P  .  p=W  .  p',  (1) ;  hence  since  P  .  j9=const. 
and  that  P  is  to  be  a  minimum,  p  must  be  a  maximum  :  but  it  is 
easy  to  see  that  p  is  a  maximum  when  it  coincides  with  the  lower 
side  of  the  cone 

*  American  edition. 

f  The  answers  given  by  Mr.  Young  to  problems  XI.  XII.  and  XIII.  are  incorrect. 

Y 


254  ELEMENTS   OF    DYNAMICS. 

.'.p=^\20^+3^\=^{i00  +  9\  =  s/{^09l, 

it  is  also  easy  to  tind  />' =0-1 96-^-2  ;  hence,  by  restoring  the  value 

of  W  and  substituting  the  values  of/),  p',  (1),  gives 

-3      0-196x142     ,  _^       ^ 

P= — ==1-377  cwt. 

^409 

acting  perpendicular  to  the  lower  side  of  the  cone. 

Note. — Since  averticalline  through  the  edge  passes  between  the 
centre  of  gravity  and  vertex  of  the  cone,  P  acts  towards  the  hori- 
zon ;  in  the  contrary  case  it  must  act  in  an  opposite  direction. 

Problem  XIII. — See  fig.  58,  p.  98.  Let  ABC  denote  the  seg- 
ment sought,  having  G  for  its  centre  of  gravity  and  GP  for  a  ver- 
tical line  passing  through  it.  Then  by  what  was  shown  at  p.  72  when 
the  segment  is  at  rest,  GP  must  pass  through  the  point  of  contact 
P  of  the  segment  with  the  horizontal  plane  RP.  Now  put  a=half 
the  transverse  axis,  and  AX=x,  then  by  prob.  X,  p.  68, 
_  Sax  —  Sx'^ 

but  since  P  is  to  coincide  with  A  (per  prob.)  we  have  AG=GP, .-. 

Any Sx^ 

the  normal  GP  =  —-^ —  ;  but  see  Young^s  Analytical  Geom. 

(edition  by  Williams,  just  published  by  Carey,  Lea  &  Blanchard, 
Philadelphia,)  p.  135,  we  have  GP(=AG)=|  a.  Hence,  by  com- 
paring the  values  of  GP  we  have  a  =  — ,  or  by  reduction 

o  CI  — ^  OC 

x^ — Sax^ — a^, .-.  by  quadratics     x=  —  (3  —  ^5)  as  required. 


Problem  XIV. — Let  normals  to  the  wall  and  inclined  plane  be 
drawn  through  the  extremities  of  the  beam ;  then  when  the  beam  is 
at  rest  they  will  evidently  intersect  each  other  in  a  vertical  line 
which  passes  through  the  centre  of  gravity  of  the  beam,  for  the  re- 
sultant of  — P, — T, which  denote  the  reactions  of  the  wall  and 
plane  must  equal  W  and  act  directly  opposite  to  it.  Let  the  line  of 
common  section  of  the  plane  of  the  normals  and  horizon  be  taken 
for  the  axis  of  x.  and  a  vertical  line  through  the  extremity  of  the 
beam  which  is  on  the  inclined  plane  be  taken  for  the  axis  of  y  ;  now 
it  is  evident  that  the  reactions  of  the  plane  and  wall  act  in  the  di- 
rections of  their  respective  normals,  .-.  put  z=the  angle  made  by 
the  normal  to  the  inclined  plane  with  the  horizon,  and  we  shall  have 
—  T  cos.  z-|-P=0,  (1)  ;  — T  sin.  z  +  W=0,  (2)  for  the  resultant 
of  the  forces  which  act  on  the  beam  when  resolved  in  the  direc- 
tionsof  the  axes  of  x  and  y  respectively,  supposing  the  beam  to 


SOLUTIONS   AND    NOTES.  255 

be  kept  at  rest  by  the  forces  which  affect  it,  (see  art.  19,  p.  24.) 
Again,  since  the  beam  is  uniform,  its  centre  of  gravity  is  at  its 
middle  point,  and  hence,  we  shall  evidently  have,  tan.  z=2  tan.  i, 

W 

(3)  ;  by  (1)  and  (2)  we  have,  tan.  z  =  — -,  T  =  ^  f  W^  +  P^  |  ; 

hence,  and  by  (3),  P  =  -^.;  T  =  ^(^^"-^  ^'+^' ^"^^LJ)  .  W 
2  tan.  t  sm.  i 

as  required. 

Problem  XV. — Let  /  denote  the  length  of  the  ladder,  rf  =  the  dis- 
tance which  the  man  has  ascended  on  it,  and  d'  =  the  distance  of 
the  common  centre  of  gravity  of  the  ladder  and  man  from  the  lower 
end  of  the  ladder  measured  on  it,  then  by  the  nature  of  the  centre 
of  gravity  (art  40,  p.  60,)  we  have 

„       i  I  w  +  dW     .,, 

now  (art.  36,  p.  50,)  the  resultant  of  w  and  W  =  lo  -f-  W  ,  and  it 
passes  through  their  common  centre  of  gravity,  (or  at  the  distance 
d'  from  the  lower  end  of  the  ladder,)  and  is  vertical  to  the  horizon. 
Let  a  straight  line  be  draAvn  from  the  lower  end  of  the  ladder  to 
the  point  of  intersection  of  a  normal  to  the  wall  at  the  upper  end 
of  the  ladder,  and  th6  vertical  through  the  common  centre  of  gra- 
vity of  w  and  W ;  then  imagine  a  plane  to  be  drawn  at  right  an- 
gles to  the  line  (thus  drawn)  through  the  lower  end  of  the  ladder ; 
hence  we  may  consider  the  ladder  as  resting  on  the  wall  and 
plane,  like  the  beam  in  the  last  problem  ;  hence,  by  using  the 
same  notation  and  proceeding  in  the  same  manner  as  in  the  last 
problem,  we  have, 

tan.  z  =  -^,  (2);  T  =  ^  (  W'^  +  P^)  .  (3). 

By  putting  ^^  +  W  =  W  ;  we  also  evidently  have 

/ 

tan.  z  =  —r-  X  tan.  i,  (4), 
a 

hence  by  (2)  and  (4)  we  have 

d'^ 
^,  W'  ^/(sm.''^-f-^cos.«^) 

as  required;  where  P^the  pressure  against  the  wall,  and  T=  the 
thrust  at  the  bottom  of  tlie  ladder. 


256  SOLUTIONS    AND    NOTES. 


Solution  of  Question  VII.  proposed  at  page  211. 

Let  i,  r^,  r^,  g,  w,  wk^,  denote  the  same  things  as  in  the  author's 
solution;  then  the  accelerating  force  of  gravity  down  the  plane 
=  o-  sin  i.  Let  us  suppose  that  the  plane  is  perfectly  smooth,  and 
that  the  wheel  is  acted  upon  by  the  motive  force  wY  at  the  point  of 
contact  P'  towards  D  ;  put  ?=the  time  from  the  origin  of  the  mo- 
tion, a:=the  space  described  by  the  centre  down  the  plane,  and  e  = 
the  angle  described  by  the  wheel  around  its  axis.  Then  by  (p.  202) 
the  centre  moves  in  the  same  manner  that  it  would  do  if  the  forces 
were   immediately  applied  to  it  without  changing  their  direction 

.•.  g  s'm.i — F=--7Y'  (0'  ^^^^  ^^^^  wheel  turns  in  the  same  manner 

that  it  would  do  if  the  centre  were  absolutely  fixed,  but  the  force 

impressed  to  turn  the  wheel  is  ivF  whose  effect  evidently=tt'Fr^, 

d^o  d^d 

and  the  effect  produced  =^-Sr^rfm=-^—i^Z;^(r=the  distance  of  any 

rfr  dt^ 

element  of  the  wheel  from  its  centre);  hence  by  (art.  154,  p.  184,) 
^.F=A:^g,  (2)  .-.  by  (1)  r,g  sin.z=r,^+i^^^  ....  (3). 

Multiply  (3)  by  r^,  then  if  we  suppose  r^e=nx,   (4),  (where 

d^O        d^x 
n=a  constant  number),  we  shall  have  r^--r-=  n—^,  and  (3)  will 

.      d^x_  r/g- sini       dx_  r^gsinA  _  r/g- sini  t^ 

^'''"^  ^~  77"-H^'  •■•  dt~  u'+nk^  '  ^  ^'  ^  ~  ?•/  -fTiP  "2'  ^^^' 
where  x,  e,  t,  are  supposed  to  commence  together,  whence  all  the 
circumstances  of  the  m.otion  become  known.  If  n  =  1,  the  wheel 
rolls  without  sliding,  the  force  wY  performing  the  office  of  the 
friction  supposed  by  the  author  in  the  case  which  he  has  considered, 
//fl  (11*  ii  y* 

If  n  is  >1,  then  ^2^.  =  ^"5;=  *^^  velocity  of  rotation  is  >  -^  = 

the  velocity  of  the  centre  =  the  velocity  of  translation,  yet  the 
wheel  does  not  move  up  the  plane  as  the  author  says  it  will,  near 
the  end  of  his  solution ;  hence  his  remarks  are  incorrect. 


SOLUTIONS    AND    NOTES.  267 


Solution  of  Question  IV.  proposed  at  page  238. 

Suppose  (with  the  author),  that  B  moves  with  the  given  velocity 

V  towards  D,  then  the  space  whicii  it  will  describe  in  the  time  t 

(from  the  origin  of  the   motion),  =  vt ;  put  6  =  the  angle  ABC, 

(e'=  its  value  at  the  origin),  r  =  the  distance  of  any  element  dm, 

of  the  rod  from  the  end  B,  M  =  the  mass  of  the  rod,  R  =  BG, 

g  =  32.  2ft.  (=  the  accelerating  force  of  gravity),  dt  =  the  constant 

element  of  the  time,  x  =  the  distance  of  dm  (estimated  on  CD), 

from  the  initial  position  of  B,  then  evidently  x  =  vt  —  r  cos.^,  (1). 

rd^d 
Now  g  cos.edt  dm  =  the  force  impressed  on  d7n,  and r— =  the 

force  received  by  it  (in  the  instant  dt)  in  a  direction  perpendicular 
to  the  rod,  (supposing  the  forces  tend  to  diminish  e)  ;  .♦.  the  motion 

lost  by  dm,  =  dm  (-j- — \- g cos.edt  j  and  the  momentary  effect  of 

this  force  to  turn  the  rod  about  B  (by  the  principle  of  the  lever), 

d^d 
=  dmr  (^-77--\-  g cos.edt),  hence  by  art.  (154)  we  shall  have  by 

using  S  as  the  sign  of  integration  relative  to  the  mass  of  the  rod, 

d^O  d^d 

Sdmr(r-j-  +  g cos.edt)  =0,  or  -r^  Sr^dm  +  g  cos.eSrdm  =  0, 

but  by  art.  (42),  (155)  Sn/m  =  RM,  Sr''dm  —  —-M,  hence  by 

o 

d^e       32" 
substitution  and  reduction, we  have  -77^+  -^cos.e  =  0,  (2),  multiply 

de^      3a" 
^2)  by  de  and  take  the  integral  and  we  have-j--| — igsin.e  :=  c  = 

dt       2R 

contrast.  (3). 

Suppose  that  at  the  origin  tJj8  velocity  V  was  impressed  on  the 
centre  of  gravity  G  of  the  rod  in  the  direction  GD',  then  by  chang- 
ing r  in  (1)  into  R  we  have  x  =  vt  —  R  cos. 9  (4)  for  the  value  of 
X  which  corresponds   to   the    centre   of  gravity,   hence  evidently 

dx     ,^  „    .      ,d9'        ,         ...      ,  .  ,     .      de'^     ,  V — V  ,„ 

-=V=t;  +  Rsm..  ^(at  the  ongm), which gives-^=(j^-^)» 

but  (3)  at  the  origin  gives 

_+  _  sm..'=  c,  hence  (^^^)  +  2|sm..'=c, 

hence,  and  by  (3)  dt  =——-——-— (5) ; 

-Jl^-T — ;)+-^(sin.e'— sin.0) 
>IVRsin.6i7     2R^  ' 

2  Y 


268  SOLUTIONS    AND   NOTES. 

the  integral  of  (5)  will  give  t  in  a  function  of  6  and  known  quanti- 
ties, .•.  reciprocally  6  will  be  found  in  a  function  of  t  and  known 
quantities,  hence  x  as  given  by  (4)  will  be  exhibited  in  terms  of  t 
and  known  quantities,  •••  we  have  the  value  of  a:  which  corresponds 
to  the  position  of  the  centre  of  gravity  at  any  time,  {t)  and  by  put- 
ting r  =  0  in  (1)  we  have  x=:  vt  which  gives  the  position  of  B  at 
the  same  time,  and  the  position  of  the  rod  is  determined  in  all  re- 
spects ;  and  it  may  be  observed  that  the  sign  +  is  to  be  used  in  (5) 
before  de  when  V  is  greater  than  v,  but  the  sign  —  in  the  contrary 
case. 

Again,  if  the  second  term  of  the  quantity  under  the  radical  in  (5) 
is  much  smaller  than  the  first,  the  integral  may  be  readily  found  in 
a  rapidly  converging  series ;  but  if  the  second  term  is  infinitely 
small  relative  to  the  first  term,  it  may  be  neglected,  and  we  shall  have 

V V 

(when  V  is  >  v),  .5 — : — ,dt  =  de  and  by  integration,  and  correction 

V  —  V 

9=i9'-{-:^—. ■/,  which  will  also  give  the  value  of  e  when  v  is  >V 

these  are  the  results  which  the  author  should  have  found  in  the  case 

(of  the  general  problem),  which  he  appears  to  have  considered  ;  and 

it  is  easy  ta  see  that  every  thing  else  required  in  this  case  is  found. 

Note. — It  is  easy  to  see  by  (1)  that  the  point  B  moves   always 

with  the  velocity  v  ;  for  it  gives  ^=  v  -\-  r  sin  e-^- ,  but  at  the  point 

dx 
B,  r=0,  hence  -^=  the  velocity  of  B  =  r,  as  it  ought  to  do. 


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